Quadrilaterals - Class 9 Maths - Extra Questions

Can a quadrilateral ABCD be a parallelogram for the following condition?
(Enter 1 for True and 0 for False)
$$AB = DC = 8$$ cm, $$AD = 4$$ cm and $$BC = 4.4$$ cm.



Can a quadrilateral ABCD be a parallelogram for the following condition?
(Enter 1 for True and 0 for False)
$$\angle A = 70^{0}\ and \ \angle C = 65^{0}$$.



If two adjacent angles of a parallelogram are in the ratio of $$2 : 3$$ find all the angles of the parallelogram.



The angles of a quadrilateral are of the measures $$120$$$$^o$$, $$90$$$$^o$$, $$72$$$$^o$$ and $$x$$$$^o$$, then find $$x$$.



Three angles of a quadrilateral are of the measure 130$$^o$$, 82$$^o$$, 40$$^o$$. Find the measure of the fourth angle.



The angles of a quadrilateral are in the ratio $$2 : 3 : 5 : 8$$. Find the smallest angle of the quadrilateral.



In square $$ABCD$$, $$\angle A = 120^o$$. If $$\angle B = \angle C = \angle D$$, then find the measures of $$\angle B$$.



The measures of two angles of a quadrilateral are $$55$$$$^o$$ each. The third angle is $$150$$$$^o$$. What is the measure of the fourth angle?



If all the sides of a parallelogram touches a circle, show that the parallelogram is a rhombus.
328273_ba31ade54a0d43178a3710a0d62c16ff.bmp



Examine whether the following points taken in order form a square.
(-1, 2), (1, 0), (3, 2) and (1, 4)



Examine whether the following points taken in order form a rectangle.
(8, 3), (0, -1), (-2, 3) and (6, 7)



Examine whether the following points taken in order form a square.
(0, -1), (2, 1), (0, 3) and (-2, 1)



Examine whether the following points taken in order form a square.
(5, 2), (1, 5), (-2, 1) and (2, -2)



A piece of wire is $$60\ cm$$ long. What will be the length of each side if the wire is used to form a square?



Three angles of quadrilateral are respectively equal to $${110^ \circ },{50^ \circ },{40^ \circ }$$. Find its fourth angle.



In the given figure, In $$\Delta PQR\quad XY\ \parallel QR$$, $$PX : QX=1 : 3$$ $$YR=4.5 cm$$ and $$QR=9 cm$$. Find  $$XY$$. 
Further if $$Ar (\Delta PXY$$)=$$A { cm }^{ 2 }$$, find in terms of $$A$$ the area of $$\Delta PQR$$ and the area of trapezium $$XYRQ$$.

867559_894a7c15cbc54f8a9fe0ecfab48458b1.png



Quadrilateral $$MORE$$
$$MO=6\ cm$$
$$OR=4.5\ cm$$
$$\angle M={60}^{o}$$
$$\angle O={105}^{o}$$
$$\angle R={105}^{o}$$



A figure is said to be regular if its sides are equal in length and angles are equal in measure. Can you identify the regular quadrilateral?



Fill in the blanks using the correct word given in brackets:
All squares are .......... (similar, congruent).



A quadrant has its all four interior angles equal. What is the measure of each interior angle.



The measure of two adjacent angles of a quadrilateral are $$110^o$$ and $$50^o$$ and the other two angles are equal. Find the measure of each angle.



In the given figures, $$PQRS$$ is a parallelogram .Find the value of $$x,y$$ and $$z$$.
1056709_1d2acb19bfc447fd9c4c1e1ad6f56f74.png



In the given figure, ABCD is a rhombus where $$BO = 12cm$$ and $$OC=5cm$$. find the values of $$x,y$$ and $$z$$
1081428_b64fac99e46a46ac8902022a5f39f8fb.png



The angle of a quadrilateral are respectively $$100^{o}, 98^{o}, 92^{o}$$. Find the fourth angle.



Show that the point $$(5, -1, 1)(7, -4, 7) , (1, -6, 10)(-1, -3, 4)$$ are the vertices of a rhombus.



Angles of a quadrilateral are in the ratio $$3:4:4:7$$ . Find all the angles of the quadrilateral.



Prove that the sum of angles of a quadrilateral is $${360}^{o}$$



The sum of two opposite angles of a quadrilateral is $${172}^{o}$$. The other two angles of the quadrilateral are equal. Find the equal angles.



In a quadrilateral if opposite vertices are joint it is called_____ 



One of the angle of quadrilatral has measure $$\dfrac{2\pi}{9}$$ radians and other three angles are in the ratio $$3: 5: 8$$, Find the measures in radian.



Find the area of trapezium whose sum of parallel sides is 28 cm and height is 10 cm.



Diagonals of a quadrilateral JOSH intersect at P in such a way that ar(JPH) = ar(OPS).Prove that JOSH is a trapezium.



The four angles of a quadrilateral are equal. Find each of them.



A rectangular park is 45 m long and 30 m wide. A path 2.5 m wide is constructed outside the park. Find the area of the path.
1222249_abd18e64beb54a488a077962594b7029.png



If the measures of internal angles of a quadrilateral are in the ratio of $$3:6:4:5$$. what are the measures of its angle?



$$ { ABCD }$$ is a parallelogram where $${ A } ( x , y ) ,  { B } ( 5,8 ) ,{ C } ( 4,7 )$$ and $${ D } ( 2 , - 4 )$$ . Find
(i) the coordinates of $$A$$.
(ii) the equation of the diagonal $$BD$$.



Find the value of $$x$$


1205748_09d294c38250495f9eb2921818aef530.jpg



Find the area and perimeter of the rectangle in the figure.
1299700_341eedac47d841c48c9b28d337f7c973.jpg



Find the length of a diagonal of a quadrilateral whose area is $$195\ m^{2}$$ and the lengths of the perpendicular to the diagonal vertices are $$8.5\ m$$ and $$11\ m$$ respectively.



Write each properties of trapezium, parallelogram, rectangle, rhombus, square.



If the four angles of a quadrilateral are equal, what will be the measure of each angle ? What is the special name of this figure ?



The ratio of the angles of a quadrilateral is  $$3 : 5 : 7 : 9.$$  Find the angles.



Find x in the picture:  
1299581_0f499eba60b140bb8450b3473243abca.PNG



Prove that a cyclic parallelogram is rectangle.



In the figure , 0 is the center of the circle and $$BA = AC$$.If$$\angle ABC = {50^.}$$,find$$\angle BOC$$ and $$\angle BDC$$
1298962_4d02a463351b40c8a43d4d8c41fb6de3.PNG



The angles of a quadrilateral are in the ratio $$2:5:5:6$$. Find the greatest angle. 



Using information given in the adjacent figure , find the value of $$x$$. 
1305419_38278f4360294b70b9238bced1fa984a.JPG



Find the value of $$a$$ in the following figures.
1337540_c490c90fae2448bca58c775d9566a1fa.png



on a line segment AB = 4 cm, construct a square,



ABCD is a quadrilateral such that diagonal AC bisects the angles $$\angle A$$ an $$\angle C$$. Prove that $$AB=AD$$ and $$CB=CD$$.



$$AO$$ and $$DO$$ are the bisectors of $$\angle A$$ and $$\angle D$$ of the quadrilateral $$ABCD$$. Prove that $$\angle AOD=\dfrac {1}{2}(\angle B+\angle C)$$



Find the value of $$x$$ in each of the following figures.
1337369_45ba7aa454e249a1851d3d82d5c2afb8.png



What is the name given to a parallelogram whose all sides are equal ?



In $$\Box ABCD$$, side $$BC\parallel  \ AD$$. Diagonal $$AC$$ and diagonal $$BD$$ Intersect in point $$Q$$. If $$AB=CD$$ .Then show that $$\Delta ABC$$ is congruent to $$\Delta DBC$$
1334130_c6d654a9141149cabb1afa44f474d114.png



In the following figure,
If $$\angle BAD=96^{o}$$, find $$\angle BCD$$ and $$\angle BFE$$
1332020_f91bd7e41c2e439b9f5eb0c367e29747.png



In $$\Box ABCD$$, side $$BC <$$ side $$ AD$$
side $$BC \parallel $$ side $$AD$$ and if
side $$BA \cong$$ side $$CD$$
then prove that $$\angle ABC \cong \angle DCB$$.
1347077_ab427e2046cb4edf97faaa460f90e4a7.png



Find area of trapezium whose parallel sides $$9cm$$ and $$5cm$$ respectively and the distance between these sides is $$8cm$$.



The angles of a qudarilateral are in the ratio 3 : 5 : 9 :Find all the angles of the quadrilateral.



Fill in the blanks: 
(i) A quadrilateral with all sides and all angles equal is a .............
(ii) A quadrilateral with four equal sides and no right angles can be called a...................
(iii) A quadrilateral with exactly two sides parallel is a......................
(iv) The diagonals of this quadrilateral are equal but not perpendicular.  The quadrilateral is a.............
(v) All rectangles, squares and rhombus are ............,but a trapezium is not. 



What is trapezium ?



The angles of quadrilateral rae $$x,x+20,x+30,x+60$$ Find $$x$$



.Use the information given in the following figure to find : 
$$(1)$$ x
$$(2)$$ $$\angle B$$ and $$\angle C$$

1355399_c236ead0bc9c4c2bb663a66545f15b5c.png



EFGH  is a parallelogram with diagonals EG and FH meeting at a point X. Show that $$ar(\Delta GXF)=ar(\Delta EXH)=ar(\Delta GXH)=ar(\Delta XEF)$$



The angles of a quadrilateral are in the ratio of $$2:3:4:5$$, then find the angles.



In quadrilateral PQRS.
$$\angle P:\angle Q:\angle R:\angle S=3:5:9:7$$. Find all the angles of quadrilateral PQRS.



If a triangle and a parallelogram are on the same base and between the same parallels, then  prove that area of triangle is half the area of parallelogram. 



Give the difference between a square and a rhombus.



2 angles of quadrilateral are $${ 70 }^{ o }$$ and $${ 80 }^{ o }$$ third angle is supplementary of $${ 70 }^{ o }$$ find the $${ 4 }^{ th }$$ angle.



Do the angles $$120,55,150,35$$ make a qudrilateral give reasons.



Find the area of the rectangles whose sides are:
$$3\ cm$$ and $$4\ cm$$ 



The angles of quadrilateral are $$x,2x,2x,5x+10$$ find the value of $$x$$



 (i) How many pairs of adjacent sides does a quadrilateral have?
(ii) How many pairs of opposite sides does a quadrilateral have?
(iii) How many pairs of adjacent angles does a quadrilateral have?
(iv) How many pairs of opposite angles does a quadrilateral have?
(v) How many diagonals does a quadrilateral have?



In the figure, given below, $$ABCD$$ is a cyclic quadrilateral in which $$\angle BAD=75^{o}; \angle ABD=58^{o}$$ and $$\angle ADC=77^{o}$$. Find: 
$$\angle BDC$$
1400717_f9a4f683d34344138fffcd337220791c.png



The angles in quadrilateral are $$2x,3x-50,4x,5x-10$$ find $$x$$



Show that sum of all sides of a quadrilateral is greater than the sum of the diagonals



ABCD is a quadrilaterial Prove that $$ A B+B C+C D+D A>A C+B D ? $$



The angle of a quadrilateral are in the ratio 3:5:9:Find all the angles of the quadrilateral.



Angles of a quadrilateral are $$\left( 4x \right) ^{ \circ  },5\left( x+2 \right) ^{ \circ  },\left( 7x-20 \right) ^{ \circ  }$$ and $$6\left( x+3 \right) ^{ \circ  }.$$ Find each angle of the quadrilateral.



The diagram shows a regular pentagon and a kite.
Complete the following statement.
The kite has rotational symmetry of order _________.

1645251_a1f6674bcb504ddea4fae4749b082ce7.png



Find the value of $$y$$
1647033_767f740553924967ab553676cf11e660.png



Find the area of the rectangles whose sides are:
$$12\ cm$$ and $$21\ cm$$ 



Find the area of the rectangles whose sides are:
$$2\ cm$$ and $$3\ cm$$ 



In the given figure 
$$b=2a+15^{\circ}$$ and 
$$c=3a+5^{\circ}$$, find the values of b and c.


1586602_cd94de4ea16f4a9091f995a3d23ea042.PNG



The angles of quadrilateral are $$3x,4x,5x,6x.$$ Find the value of $$x.$$  



Angles of a quadrilateral are $$\left( 4x \right) ^{ \circ  },5\left( x+2 \right) ^{ \circ  },\left( 7x-20 \right) ^{ \circ  }$$ and $$6\left( x+3 \right) ^{ \circ  }.$$ Find
the value of $$x$$.



The sum of the angles of a quadrilateral is_______



A quadrilateral has ...... diagonals.



The tree angles of a quadrilateral are $$ 76^o, 54^o $$ and $$ 108^o $$. Find the measure of the fourth angle.



Solve the following :
Of the four quadrilaterals - square , rectangle , rhombus and trapezium - one is somewhat different from the others because of its design. Find it and give justifications.



In the given figure, $$ABCD$$ is a quadrilateral shaped field in which diagonal $$BD$$ is $$36\ m$$, $$AL \bot  BD$$ and $$CM \bot BD$$ such that $$AL=19\ m$$ and $$CM=11 \ m$$. Find the area

1774814_6f6566c788f94e079542afc8af0702e5.png



Solve the following :
The point of intersection of diagonals of a quadrilateral divides one diagonal in the ratio $$ 1 : 2 $$. Can it be a parallelogram ? Why or why not ?



A quadrilateral has_______ angles.



Prove that the sum of the angles of a quadrilateral is $$ 360^o $$



The figure given below, shows a trapezium $$ABCD. M$$ and $$N$$ are the mid-points of the non-parallel sides $$AD$$ and $$BC$$ respectively. Find:
$$MN$$, if $$AB=11\ cm$$ and $$DC=8\ cm$$
1840883_b94a152fe988425eacc6b8e581187e56.png



The diagonal of a quadrilateral intersect at right angles. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.



Prove that the figure obtained by joining the mid-point of the adjacent sides of a rectangle is a rhombus.



In triangle $$ABC, M$$ is mid-point of $$AB$$ and a straight line through $$M$$ and parallel to $$BC$$ cuts $$AC$$ at $$N$$. Find the lengths of $$AN$$ and $$MN$$, if $$BC=7\ cm$$ and $$AC=5\ cm$$
1840868_1f3b555a245746b2a5c280b4bf7f8251.png



Solve the following :
Two sticks each of length $$ 5\,\text{cm} $$ are crossing each other such that they bisect each other. What shape is formed by joining their endpoints? Give reason.



D, E and F are mid-point of the sides BC, CA and AB respectively of a $$ \triangle $$ ABC. Prove that:
(i) FDCE is a parallelogram
(ii) area of $$ \Delta $$ DEF $$ =1 / 4 $$ area of $$ \Delta \mathrm{ABC} $$
(iii) area of $$ \| \mathrm{gm} $$ FDCE $$ =1 / 2 $$ area of $$ \Delta \mathrm{ABC} $$



In the quadrilateral $$ABCD$$; prove that
$$AB+BC+CD > DA$$



The figure given below, shows a trapezium $$ABCD. M$$ and $$N$$ are the mid-points of the non-parallel sides $$AD$$ and $$BC$$ respectively. Find:
$$AB$$, if $$DC=20\ cm$$ and $$MN=27\ cm$$
1840885_45f3e2fdc47a4e1dac124cc33e9acff5.png



The figure given below, shows a trapezium $$ABCD. M$$ and $$N$$ are the mid-points of the non-parallel sides $$AD$$ and $$BC$$ respectively. Find:
$$DC$$, if $$MN=15\ cm$$ and $$AB=23\ cm$$
1840887_6cac4814cf554c6f981484e96b67a654.png



The following figure shows a trapezium $$ABCD$$ in which $$AB\parallel DC. P$$ is the mid-point of $$AD$$ and $$PR\parallel AB$$. Prove that:
$$PR=\dfrac{1}{2}(AB+CD)$$
1840876_fd3dd21175024cca8a7721ce57c84ced.png



$$D, E$$ and $$F$$ are the mid-point of the sides $$AB, BC$$ and $$CA$$ respectively of triangle $$ABC, AE$$ meets $$DF$$ at $$O. P$$ and $$Q$$ are the mid-points of $$OB$$ and $$OC$$ respectively. Prove that $$DPQF$$ is a parallelogram.



A parallelogram $$ABCD$$ has $$P$$ the mid-point of $$DC$$ and $$Q$$ a point of $$AC$$ such that $$CQ=\dfrac{1}{4}AC. PQ$$ produced meets $$BC$$ at $$R$$
Prove that :
$$R$$ is the mid-point of $$BC$$.
1840916_9b2b24a20ab2479794bcc069ce223201.png



In trapezium $$ABCD, AB$$ in parallel to $$DC; P$$ and $$Q$$ the mid-points of $$AD$$ and $$BC$$ respectively. $$BP$$ produced meets $$CD$$ produced at point $$E$$. Prove that:
Point $$P$$ bisects $$BE$$
1841046_7d9aa94bc2f74e0ca20d79d402ac3daa.png



In the quadrilateral $$ABCD$$; prove that
$$AB+BC+CD + DA > 2BD$$



A parallelogram $$ABCD$$ has $$P$$ the mid-point of $$DC$$ and $$Q$$ a point of $$AC$$ such that $$CQ=\dfrac{1}{4}AC. PQ$$ produced meets $$BC$$ at $$R$$
Prove that :
$$PR=\dfrac{1}{2}DB$$
1840919_85b8dc37ca9244139d93f9ed60f92cb4.png



In trapezium $$ABCD, AB$$ in parallel to $$DC; P$$ and $$Q$$ the mid-points of $$AD$$ and $$BC$$ respectively. $$BP$$ produced meets $$CD$$ produced at point $$E$$. Prove that:
$$PQ$$ is parallel to $$AB$$.
1841049_d3eb7d04eeee47a982055e75434782d8.png



In a triangle $$ABC, P$$ is the mid-point of side $$BC$$. A line through $$P$$ and parallel to $$CA$$ meets $$AB$$ at point $$Q$$; and a line through $$Q$$ and parallel to $$BC$$ meets median $$AP$$ at point $$R$$. Prove that:
$$BC=4QR$$



In a triangle $$ABC, P$$ is the mid-point of side $$BC$$. A line through $$P$$ and parallel to $$CA$$ meets $$AB$$ at point $$Q$$; and a line through $$Q$$ and parallel to $$BC$$ meets median $$AP$$ at point $$R$$. Prove that:
$$AP=2AR$$



In the quadrilateral $$ABCD$$; prove that
$$AB+BC+CD + DA > 2AC$$



In quadrilateral $$ABCD$$, side $$AB$$ is the longest and side $$DC$$ is the shortest. Prove that:
$$\angle D > \angle B$$



$$D$$ and $$F$$ are mid-points of sides $$AB$$ and $$AC$$ of a triangle $$ABC$$. A line through $$F$$ and parallel to $$AB$$ meets $$BC$$ at point $$E$$.
Prove that $$BDFE$$ is a parallelogram.
1841051_2da0abc2da9c442fa16ef89fd4be8d03.png



In triangle $$ABC$$, and $$B$$ is obtuse. $$D$$ and $$E$$ are mid-point of sides $$AB$$ and $$BC$$ respectively and $$F$$ is a point on side $$AC$$ such that $$EF$$ is parallel to $$AB$$. Show that $$BEFD$$ is a parallelogram.
1841121_a9b39f1f898d493da51f08536081a50e.png



In parallelogram $$ABCD, E$$ and $$F$$ are mid-points of the sides $$AB$$ and $$CD$$ respectively. The line segments $$AF$$ and $$BF$$ meet the line segments $$ED$$ and $$EC$$ at points $$G$$ and $$H$$ respectively.
Prove that:
$$GEHF$$ is a parallelogram.
1841131_f38f502f41484f98968b28c22964e6c9.png



$$D$$ and $$F$$ are mid-points of sides $$AB$$ and $$AC$$ of a triangle $$ABC$$. A line through $$F$$ and parallel to $$AB$$ meets $$BC$$ at point $$E$$.
Find $$AB$$, if $$EF=4.8\ cm$$.



In a triangle $$ABC, AD$$ is a median and $$E$$ is mid-point of median $$AD$$. A line through $$B$$ and $$E$$ meets $$AC$$ at point $$F$$.
Prove that: $$AC=3AF$$



In the figure, given below, $$2AD=AB, P$$ is mid-point of $$AB, Q$$ is mid-point of $$DR$$ and $$PR\parallel BS$$. Prove that:
$$AQ\parallel BS$$.
1841095_7bcf0a77ce214dcfbd2a41d13b822c2f.png



The side $$AC$$ of a triangle $$ABC$$ is produced to point $$E$$ so that $$CE=\dfrac{1}{2}AC.\ D$$ is the mid-point of $$BC$$ and $$ED$$ produced meets $$AB$$ at $$F$$. Lines through $$D$$ and $$C$$ are drawn parallel to $$AB$$ which meet $$AC$$ at point $$R$$ respectively. Prove that:
$$3DF=EF$$
1841104_317cd2d605694c3fbdd42a635387a501.png



The side $$AC$$ of a triangle $$ABC$$ is produced to point $$E$$ so that $$CE=\dfrac{1}{2}AC.\ D$$ is the mid-point of $$BC$$ and $$ED$$ produced meets $$AB$$ at $$F$$. Lines through $$D$$ and $$C$$ are drawn parallel to $$AB$$ which meet $$AC$$ at point $$R$$ respectively. Prove that:
$$4CR=AB$$
1841105_f407a71648e346419c041e496a340164.png



In parallelogram $$ABCD, E$$ and $$F$$ are mid-points of the sides $$AB$$ and $$CD$$ respectively. The line segments $$AF$$ and $$BF$$ meet the line segments $$ED$$ and $$EC$$ at points $$G$$ and $$H$$ respectively.
Prove that:
Triangles $$HEB$$ and $$FHC$$ are congruent.
1841130_68850d29887a433194e59f8dd3b03206.png



In triangle $$ABC, M$$ is mid-point of $$AB, N$$ is mid-point of $$AC$$ and $$D$$ is any point in base $$BC$$. Use Intercept Theorem to show that $$MN$$ bisects $$AD$$.



Can the angles $$110^\circ, 80^\circ, 70^\circ and\ 95^\circ$$ be the angles of a quadrilateral? Why or why not?



If the quadrilateral formed by joining the mid-points of the adjacent sides of quadrilateral $$ABCD$$ is a rectangle, show that the diagonal $$AC$$ and $$BD$$ intersect at right angle.



In triangle $$ABC; D$$ and $$E$$ are mid-points of sides $$AB$$ and $$AC$$ respectively. Through $$E$$, a straight line is drawn parallel to $$AB$$ to meet $$BC$$ at $$F$$. Prove that $$BDEF$$ is a parallelogram
If $$AB=16\ cm, AC=12\ cm$$ and $$BC=18\ cm$$, find the perimeter of the parallelogram $$BDEF$$.
1841162_2a96be1db423401ba6f876d7d70e519d.png



In quadrilateral $$ABCD , AD = BC$$ and $$BD = CA$$ prove that 
$$ \angle ADB = \angle BCA $$
1841396_063fcd4733a841f29fbbb79c81366e78.png



In parallelogram $$ABCD, E$$ is the mid-point of $$AB$$ and $$AP$$ is parallel to $$EC$$ which meets $$DC$$ at point $$O$$ and $$BC$$ produced at $$P$$. Prove that:
$$O$$ is mid-point of $$AP$$.
1841178_54ace0064c8740b5bc670795142ad6fd.png



In quadrilateral ABCD , AD = BC and BD = CA prove that 
$$ \angle DAB = \angle CBA $$
1841402_bba3207704e44a2f99a9d09a9c68cb68.png



ABCD is a rectangle in which diagonal AC bisects A as well as C. Show that ABCD is a square.



A quadrilateral ABCD will be a parallelogram for the following conditions.
$$\angle D + \angle B = 180^{0}$$
If true then enter $$1$$ and if false then enter $$0$$



$$ABCD$$ is a convex quadrilateral with
$$AB + BC \leq  AC + CD$$. Prove that $$AB < AC$$.



If a quadrilateral ABCD is such that $$\displaystyle \vec{AB}= b, \vec{AD}= d$$ and $$\displaystyle \vec{AC}= pb+qd(p+q\geq 1),$$ then show that the area of the quadrilateral is $$\displaystyle \frac{1}{2}\left ( p+q \right )\left | b\times d \right |.$$



If all angles of a quadrilateral are congruent, then find their measures.



Use the information given in the following figure to find :
$$x$$ (in degrees).

179860.jpg



The bisectors of angles $$B$$ and $$C$$ of a parallelogram $$ABCD$$ meet at point $$O.$$ If the triangle $$OBC$$ formed is an isosceles right triangle, then $$ABCD$$ is a rectangle.
If the above statement is true then mention the answer as $$1,$$ else mention $$0$$ if false.



State True or False: 
The figure obtained by joining the mid-points of the adjacent sides of a rectangle is a Trapezium



Construct a rhombus whose diagonals are $$47\ cm$$ and $$54\ cm$$.



In rectangle ABCD, the bisectors of angles B and C meet at point O then triangle OBC is an isosceles right triangles.
If the above statement is true then mention answer as 1, else mention 0 if false



In the fig. given measure of angle $$\angle PLN=110^o$$. Then the measure of $$\angle LMN$$ in degrees is:
194598_01dd72a8bc084c00b71d8912c81f8363.png



$$ABCD$$ is a trapezium with $$AB\: // \:DC$$ . A line parallel to $$AC$$ intersects $$AB$$ at point $$M$$ and $$BC$$ at point $$N$$ . Prove that : $$area \: of \: \bigtriangleup ADM \: = \: area \: of \: \bigtriangleup \: ACN$$.



The diagonal BD of a parallelogram ABCD bisects angles B and D. Prove that ABCD is a rhombus.



Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a rectangle is a rhombus.
283237.png



The angles of a quadrilateral $$ABCD$$ are in the ratio $$4 : 9 : 10: 13$$. Find the angles.



In the adjoining figure, $$ABCD$$ is a trapezium in which $$AB\;\parallel\;DC\;and\;AD=BC$$. If $$P,\,Q,\,R,\,S$$ be respectively the midpoints of $$BA,\,BD\;and\;CD,\;CA$$ then show that $$PQRS$$ is a rhombus.
327591_12bc5e0522394df68338f1c3fab47b5c.bmp



In the given figure, find the values of $$x$$ and $$y$$.
326576_3c17f841f2e542f48785977272d3b5af.png



Prove that all four angles of a quadrilateral cannot be obtuse angles.



Prove that all four angles of a quadrilateral cannot be acute angles



In Fig. 6.43, if $$PQ \perp PS, PQ\parallel SR,\angle SQR={ 28 }^{ 0}$$ and $$\angle QRT={ 65 }^{ 0}$$, then find the values of $$x$$ and $$y$$.
463685.png



The following figures GUNS and RUNS are parallelograms. Find $$x$$ and $$y$$. (Lengths are in cm)

463376_7d12363a5e494d96a65f3d66e3b6d41a.png



The above figure is a parallelogram. Find the angle measures $$x, y$$ and $$z$$. State the properties you use to find them.

463375.png



The angles of quadrilateral are in the ratio $$3 : 5 : 9 : 13$$. Find all the angles of the quadrilateral.




Consider the following parallelograms. Find the values of the unknowns $$x, y, z$$
463370_4f003dd39fb24715a333f540ff334c8b.png



Given a parallelogram $$ABCD$$. Complete each statement along with the definition or property used.
(i) $$AD =$$
(ii) $$\angle DCB =$$
(iii) $$OC =$$
(iv) $$m \angle DAB + m \angle CDA =$$

463369.PNG



Can a quadrilateral $$ABCD$$ be a parallelogram if
(i) $$\angle D + \angle B = 180^{\circ}$$? 
(ii) $$AB = DC = 8$$ cm, $$AD = 4$$ cm and $$BC = 4.4$$ cm? 
(iii) $$\angle A = 70^{\circ}$$ and $$\angle C = 65^{\circ}$$?



In the above figure both RISK and CLUE are parallelograms. Find the value of $$x$$.
463379_e50bd44b788c49c0841283d2239f04fe.png



Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other



In parallelogram $$ABCD$$, two point $$P$$ and $$Q$$ are taken on diagonal $$BD$$ such that $$DP = BQ$$. Show that
(i) $$\triangle APD\cong \triangle CQB$$
(ii) $$AP = CQ$$
(iii) $$\triangle AQB \cong \triangle CPD$$
(iv) $$AQ = CP$$
(v) $$APCQ$$ is a parallelogram
463883.jpg



$$ABC$$ is a triangle right angled at $$C$$. A line through the mid-point $$M$$ of hypotenuse $$AB$$ and parallel to $$BC$$ intersects $$AC$$ and $$D$$. Show that
(i) $$D$$ is the mid-point of $$AC$$
(ii) $$MB\perp AC$$
(iii) $$CM = MA = \dfrac {1}{2}AB$$



$$ABCD$$ is a trapezium in which $$AB \parallel DC, BD$$ is a diagonal and $$E$$ is the mid-point of $$AD$$. A line is drawn through $$E$$ parallel to $$AB$$, intersecting $$BC$$ at $$F$$ (see Fig). Show that $$F$$ is the mid-point of $$BC$$.

463892_2f41f39ff3704fd790bc8e856fef02e5.png



Show that the diagonals of a parallelogram divide it into four triangles of equal area.



$$ABCD$$ is a parallelogram and $$AP$$ and $$CQ$$ are perpendiculars from vertices $$A$$ and $$C$$ on diagonal $$BD$$ Show that
i) $$\triangle\ APB \cong \triangle\ CQD$$
ii) $$AP = CQ$$
463884_aa0f9c0f61124b9aafe588c7cfd35a07.png



In a parallelogram $$ABCD,\ E$$ and $$F$$ are the mid-points of sides $$AB$$ and $$CD $$ respectively (see Fig). Show that the line segments $$AF$$ and $$EC$$ trisect the diagonal $$BD$$.

463893_a2f380c87d2d4fbda500a37f15f79298.png



If $$E, F, G$$ and $$H$$ are respectively the mid-points of the sides of a parallelogram $$ABCD$$, show that $$ar(EFGH) = \dfrac {1}{2} ar (ABCD)$$




In the given Figure, $$ABCD$$ is a parallelogram, $$AE\perp DC$$ and $$CF \perp AD$$. If $$AB = 16\ cm, AE = 8\ cm$$ and $$CF = 10\ cm$$, find $$AD$$.
463915_9ccf8ca241c0440fb24bf02c05f7b6fb.png



Prove that the rhombus with equal diagonals is a square.



In the figure, suppose $$\angle P$$ and $$\angle Q$$ are supplementary angles and $$\angle R = 125^{\circ}$$. Find the measures of $$\angle S$$.
558486_a6fc0eb31e134e3eb464aabd1064904e.png



All rectangles are parallelograms, but all parallelograms are not rectangles. Justify this statements



Is parallelogram a rectangle? Can you call a rectangle a parallelogram?



Two angles of a quadrilateral are $$70^{\circ}$$ and $$130^{\circ}$$ and the other two angles are equal. Find the measure of these two angles.



Complete the following table by writing (YES) if the property holds for the particular Quadrilateral and (NO) if property does not holds.
PropertiesTrapeziumParallelogramRhombusRectangleSquare
a. Consecutive angles are supplementary



Complete the following table by writing (YES) if the property holds for the particular Quadrilateral and (NO) if the property does not hold.
PropertiesTrapeziumParallelogramRhombusRectangleSquare
Diagonals are equal



Suppose in a rectangle, the diagonals are perpendicular to each other. Prove that it is a square.



Complete the following table by writing (YES) if the property holds for the particular Quadrilateral and (NO) if property does not holds.
PropertiesTrapeziumParallelogramRhombusRectangleSquare
a. Opposite angles are equal



Complete the following table by writing (YES) if the property holds for the particular Quadrilateral and (NO) if property does not holds.
PropertiesTrapeziumParallelogramRhombusRectangleSquare
a. All sides are equal



Complete the following table by writing (YES) if the property holds for the particular Quadrilateral and (NO) if property does not holds.
PropertiesTrapeziumParallelogramRhombusRectangleSquare
a. One pair of opposite sides are parallel



Suppose in a quadrilateral, the diagonals bisect the angles at the vertices. Prove that it is a rhombus.



Complete the following table by writing (YES) if the property holds for the particular Quadrilateral and (NO) if property does not holds.
PropertiesTrapeziumParallelogramRhombusRectangleSquare
a. Two pairs of opposite sides are parallel



The midpoint of the side AB of a triangle ABC is D and P is any point on BC. Suppose Q is a point on AC such that ADPQ is a parallelogram. Prove that DQ is parallel to BC.



Complete the following table by writing (YES) if the property holds for the particular Quadrilateral and (NO) if property does not holds.
PropertiesTrapeziumParallelogramRhombusRectangleSquare
a. Diagonals bisect each other



$$ABCD$$ is quadrilateral in which $$AB\parallel CD$$. If $$AD = BC$$, show that $$\angle A = \angle B$$ and $$\angle C = \angle D$$



Three vertices of a parallelogram $$\text{ABCD}$$ taken in order are $$\text{A}\left( 3,6 \right)$$, $$\text{B}\left( 5,10 \right)$$ and $$\text{C}\left( 3,2 \right) $$ find :
(i) the coordinates of the fourth vertex $$\text{D}$$.
(ii) length of diagonal $$\text{BD}$$.



Show that the line segments joining the midpoints of the opposite sides of a quadrilateral and bisect each other



$$ABCD$$ is a parallelogram $$AP$$ and $$CQ$$ are perpendicular drawn from vertices $$A$$ and $$C$$ on diagonal $$BD$$ (see figure) show that
$$AP = CQ$$
570436_0d404d4bc1bc48eea38466e6f933b859.png



The four angles of a quadrilateral are in the ratio $$1 : 2 : 3 : 4$$. Find the measure of each angle of the quadrilateral.



Complete the following table by writing (YES) if the property holds for the particular Quadrilateral and (NO) if property does not holds.
Quadrilateral nameTrapeziumParallelogramRhombusRectangleSquare
Each angle is a right angle



Complete the following table by writing (YES) if the property holds for the particular Quadrilateral and (NO) if property does not holds.
PropertiesTrapeziumParallelogramRhombusRectangleSquare
Diagonals are perpendicular to each other



The opposite angles of a parallelogram are $$(3x - 2)^{\circ}$$ and $$(x + 48)^{\circ}$$. Find the measure of each angle of the parallelogram.



$$ABCD$$ is a rectangle $$AC$$ is diagonal. Find the angles of $$\triangle ACD$$. Give reasons



Prove that a line drawn through the mid-point of one side of a triangle parallel to second side bisects the third side.



In the given figure, $$AED$$ is an isosceles triangle with $$AE = AD. ABCD$$ is a parallelogram and $$EGF$$ is a line segment. If $$\angle DCF = 65^{\circ}$$ and $$\angle EFB = 100^{\circ}$$, then the number of diagonals of a regular polygon having its each exterior angle equal to the measure of $$\angle AEG$$, is
707085_5c75dd5ab1b24bf88fbf7442f8542b16.png



If the measures of three angles of a quadrilateral are $$100^{\circ}, 84^{\circ}$$ and $$76^{\circ}$$ then, find the measure of fourth angle.



In $$\Delta ABC$$, $$AD$$, is a median and $$E$$ is the mid point of $$AD$$. If $$BE$$ is produced to meet $$AC$$ in $$F$$, show that $$AF=\dfrac { 1 }{ 3 } AC$$.



Find the four angles of a cyclic quadrilateral ABCD in which $$\angle A=(2x-1)^o$$, $$\angle B=(y+5)^o \angle C=(2y+15)^o$$ and $$\angle D =(4x-7)^o$$.



Find the angles of a rhombus whose diagonal is equal to a side. 



The perimeter of a rhombus is equal to $$48$$, and the sum of the lengths of the diagonals is equal to $$26$$. Find the area of the rhombus.



In trapezium ABCD, side AB $$\parallel$$ side DC, diagonals AC and BD intersect in point O. If AB=$$20$$, DC=$$6$$, OB=$$15$$, then find OD.
970900_a1ec650e3408411580b981da5c94e935.png



If the area of a rectangle is $$200$$ sq.m and the difference of its length and breadth is $$10$$ cm, find its length, breadth and perimeter.



In the adjoining figure, $$ABCD$$ is a square. A line segment $$CX$$ cuts $$AB$$ at $$X$$ and the diagonal $$BD$$ at $$O$$ such that $$\angle$$ $$COD = $$ $$80^0$$ and $$\angle$$ $$OXA =$$ $$x^0$$ . Find the value of $$x$$.
760262_397f8c34e25d437d8c4e844ac9497973.png



State and prove mid point theorem.




In the given figure, AB $$\parallel$$ DC Prove that(i) $$\triangle DMU \sim \triangle BMV$$(ii) $$DM \times BV = BM \times DU$$
969546_2cd9868b06944374abb3173d833c6bc3.png



The diagonals of a quadrilateral intersect at right angles. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.



If $$OC$$ and $$OD$$ are the bisectors of $$\angle C$$ and $$\angle D$$ respectively and $$\angle A =115^{\circ}$$, find $$\angle B$$.
1053010_7c9cf17b011e4d3ebb4a5d7323fd2654.png



The following figure shows a trapezium $$ABCD$$ in which $$AB$$ is parallel to $$DC$$ and $$AD=BC$$.
Prove that :
$$i)\; \;\;\angle DAB = \angle CBA$$
$$ii)\;\; \angle ADC = \angle BCD$$
$$iii) \;AC=BD$$
$$iv)\;\; OA=OB$$ and $$OC=OD$$



$$ABCD$$ is a rectangle in which diagonal AC bisects $$\angle A $$ as well as$$\angle C$$. 
Show that,
(i) $$ABCD$$ is square 
(ii) diagonal $$BD$$ bisects $$\angle B$$ as well as $$\angle D$$.  



Three angles of a quadrilateral are equal to $${100^ \circ },{60^ \circ }$$ and  $${80^ \circ }$$, respectively. Find its fourth angle.



Three angles of a quadrilateral $$ABCD$$ are $$\angle A=70^o,\angle B=50^o$$ and $$\angle C=82^o$$. Find $$\angle D$$.



In a quadrilateral ABCD, show that $$AB+BC+CD+DA>AC+BD$$.



In the adjacent figure the $$AB$$ and $$AC$$ of $$\triangle ABC$$ are produced to point $$E$$ and $$D$$ respectively If bisectors $$BO$$ and $$CO$$ of $$\angle CBD$$ and $$\angle BCD$$ respectively meet at point $$O$$, then prove that $$\angle BOC=90^{o}-\dfrac {1}{2} \angle BAC$$.
1039468_a1c9585698e04d36a05a635364285a47.png



Draw two angles of $${90^ \circ }$$ at the ends of a line segment. Extend these rays. Do you get a triangle. Join these 2 rays by another line segment which shape can yo get? What will be the measure of all the angles?



A quadrilateral $$ABCD$$ is drawn to circumscribe a circle (see Fig). prove that $$AB+CD=AD+BC$$.
1056817_af837a30ea33409aa2731e507ac38668.png



Angles of a quadrilateral are in the ratio $$1: 2: 3: 4$$. Find all angles. What special name can you give to this quadrilateral and why?



ABCD is a quadrilateral in which AD$$=$$BC and $$\angle$$ DAB$$=\angle$$ CBA. Prove that $$\angle$$ABD $$=\angle$$BAC.
1071345_43383272590d4a3aa52c5dc127243edc.png



The perimeter of a square is $$(4x+20)cm$$. What will be the length of its diagonal?



Find the measurements of the angles of quadrilateral if they are in ratio of  $$2:3:4:6$$



In $$\square$$PQRS, $$m\angle Q$$ is $$10^o$$ less, $$m\angle R$$ is $$20^o$$ less and $$m\angle S$$ is $$30^o$$ less, than $$m\angle P$$, then find measure of each angle.



In $$\Box ABCD$$, $$\bar{AB}\parallel\bar{CD}$$. Prove that $$AD=BC$$ and $$AB=CD$$.



The diagonals of a rhombus $$ABCD$$ are $$6\ cm$$ and $$8\ cm$$. Find the length of a side of the rhombus.



The measure of three angles of a quadrilateral are equal. If measure of each angle is $$95^\circ$$, then find measure of its fourth angle.



In the adjoining figure, $$D$$ and $$E$$ are respectively the midpoints of sides $$AB$$ and $$AC$$ of $$\triangle{ABC}$$. If $$PQ\parallel BC$$ and $$CDP$$ and $$BEQ$$ are straight lines, then prove that $$ar(\triangle{ABQ})=ar(\triangle{ACP})$$.



In a quadrilateral ABCD. AO and BO are bisectors of angle A and angle B respectively. Prove that $$\angle$$AOB$$=\dfrac{1}{2}\{\angle C+\angle D\}$$.



In triangle $$ABC, XY || AC$$ and divide the triangle into two parts of equal areas. Find the ratio $$\dfrac{AX}{AB}$$.



In a $$\Delta ABC$$, the internal bisectors of $$\angle B$$ and $$\angle C$$ meet at $$P$$ and the external bisectors of $$\angle B$$ and $$\angle C$$ meets at Q. Prove that $$\angle BPC + \angle BQC = {180^0}$$.



The interior angles of a quadrilateral are in the ratio $$3:5:6:4.$$ Find the angles.



If BC is a diameter of a circle of circle of centre O and OD is perpendicular to the chord $$\overline { AB } $$ of a circle then $$CA\equiv$$________
1140817_a0f3283cbc85462b9ab0d30b7eb720fa.png



Two opposite angles $$A$$ and $$C$$ of a parallelogram $$ABCD$$ are $$(6x-17)^{o}$$ and $$(x+63)^{o}$$ respectively as shown the figure. Find the measure of angle $$A$$.
1085692_1be667c37c674c93b3df40af228d850e.png



In figure given, we have $$X$$ and $$Y$$ are the midpoints of $$AB$$ and $$BC$$ and $$AX=CY$$. Show that $$AB=BC$$.
1126326_a0eebd5b78e0484b9e1569b682deae82.png



Three angles of a quadrilateral are equal. The fourth angle measure $${120^ \circ }$$. What is the measure of each of the equal angles?



In a quadrilateral ABCD, the bisector of $$\angle$$C and $$\angle$$D intersect at O. Prove that $$\angle$$COD$$=\dfrac{1}{2}(\angle A+\angle B)$$.



Show that the point  A (3,5) B (6,0) C (1,-3) D (-2,2) are the vertex of the square ABCD. 



From the given groups of angles which are possible group angles of a quadrilateral:
i) $$60^{\circ}, 70^{\circ}, 80^{\circ}$$, and $$145^{\circ}$$
ii) $$90^{\circ}, 90^{\circ}, 90^{\circ},$$ and $$90^{\circ}$$



Find the area of a rhombus if its vertices are $$(3, 0), (4, 5), (-1, 4)$$ and $$(-2, -1)$$ taken in order.



The sum of two angles of a quadrilateral is $$150^{o}$$ and the other angles are in the ratio $$2:3$$. Find the measure of two angles.



In Fig $$ABCD$$ is a rhombus in which $$\alpha ABC=56^{o}$$. Which is the measure of $$\angle ACD$$?
1151429_d5498cde72fd4c579782f5e6bc8e4216.png



In the quadrilateral $$PQRS$$,  the angles are in the ratio of $$3: 4: 5: 6$$.  Find the angles of $$PQRS$$.  What is the special name given to $$PQRS$$?



Prove that each of the four sides of a rhombus is of the same length.



What is parallelogram ?



In the adjoining figure, $$A C \perp O C \text { and } A D \perp O D$$. Find $$\angle C A D$$
1183630_46868e7b74e24ef9ab5cdcedef94b137.png



Show that the line segment joining the mid points of the opposite sides of a quadrilateral bisect each other.



Show that the points $$A(1,2),B(5,4),C(3,8)$$ and $$D(-1,6)$$ are the vertices of a square.



ABCD is a rectangle in which diagonal AC bisects $$\angle \mathrm { A }$$ as well as $$\angle \mathrm { C }$$ . Show that
(i) ABCD is a square
(ii) diagonal BD bisects $$\angle \mathrm { B }$$ as well as $$\angle \mathrm { D }$$. 



If the diagonals of a quadrilateral bisect each other, then prove that it is a parallelogram.



$$E$$ and $$F$$ are points on diagonals $$AC$$ of a parallelogram $$ABCD$$ such that $$AE=CF$$. Show that $$BFDE$$ is a parallelogram.



The four angles of a quadrilateral are equal. Draw this quadrilateral in your notebook. Find each of them.



In a quadrilateral, the angles are $$x^{o},(x+10)^{o},(x+20)^{o},(x+30)^{o}$$. Find the angles.



Construct Rhombus MATH with $$\mathrm { AT } = 4 \mathrm { cm } , \angle \mathrm { MAT } = 120 ^ { \circ }$$



If $$A , B . C$$ are $$( 3,3,3 ) , ( 0,6,3 )$$ and $$( 1,7,7 )$$ respectively. find $$D ( x , y , z )$$ such that ABCD is a square



ABCD is a trapezium is which $$AB\parallel DC$$, BD is a diagonal and E is the midpoint of AD. A lie is drawn through E parallel. AB IN intersecting BC at F. show that F is the midpoint of BC



If ABCD is a  rohmbus with $$\angle ABC={ 56 }^{ 0 }$$ Find the measure of $$\angle ADC$$



Prove that, the sum of either pair of opposite angles of a cyclic quadrilateral is 180.



In $$\triangle PQR$$,point M is on side PQ and point S is on the side PR such that QRSM is what kind of figure?



Find the lengths of diagonals $$AC$$ and $$BD$$. Given $$AB = 60\ cm$$ and $$\angle BAD = 60^{\circ}$$.
1216343_f8c264b221c04749928bf440911092dd.png



In the given figure, ABCD and PQRB are rectangles where Q is the mid point of BD. If QR=5cm, then find the length AB.
1218681_f9bb288dcdb0455d98d29630a57c09a2.png



$$AB$$ and $$CD$$ bisect each other of $$K$$.
Prove that $$AC =BD$$
1214709_e8925a60596a4f2cba7d76aa813b5adc.png



The angles of quadrilateral are in the ratio $$3:5:9:13$$. Find all the angles of the quadrilateral.



Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
1247243_d78bdad4120d405ea6f171cbb657ba6c.png



If one side of a square is represented by $$4x-7$$ and the adjacent side is represented by $$3x+5$$, find the value of $$x$$.



The four angles of a quadrilateral are in the ratio $$1 : 2 : 3 : 4$$.Find the measure of each angle of the quadrilateral.



Fill in the blanks 
If three acute angles of a quadrilateral measure $$70^0$$ each, then the measure of the fourth angle is ____ . 



Prove that the sum of all the angle of a quadrilateral is $$360^\circ.$$



One single of quadrilateral it of $${ 108 }^{ \circ  }$$ and the remaining three angles are equal. Find each of the three equal angles.



In the adjoining figure $$PQRS$$ is a rectangle. Identify the congruent triangles formed by the diagonals. 
1232470_9afe32b38dd443a59e733882b5a82a91.png



The angles of a quadrilateral are in the ratio $$2 : 4 : 5 : 7$$. Find the angles.



Show that the line segments joining the mid-points of the opposite side of a quadrilateral bisect each other.



In the given figure, ABCD is a cyclic quadrilateral , whose diagonals intersect at $$E$$  such that $$\angle BAC= 30^{0}$$ , $$\angle DBC= 70^{0}$$ and AB=BC,Find  $$\angle ACD$$.
1271709_3e3e3ed2f07e4494a2af423a51713e20.PNG



Find the unknown angle.
1274974_f887d915027c4ccea3622c28af9fdfd8.png



Show that the four triangles as shown in the adjoining fig. formed by diagonals and sides of rhombus are congruent.
1273957_3c6cc4b685a040d3ae96b355ef0d283d.png



In the given figure, $$AC$$ is the bisector of $$\angle A$$. If $$AB=AC, AD=CD$$ and $$\angle ABC=75^{o}$$, find the value of $$x$$ and $$y$$.
1263852_d54573013afe4fa487d7d5331db7eafe.png



In a trapezium  $$ABCD$$  with  $$AB$$  parallel to  $$CD,$$  the diagonals intersect at $$P.$$  The area of  $$\triangle A B P$$  is  $$72\mathrm { cm } ^ { 2 }$$  and of  $$\Delta C D P$$  is  $$50\mathrm { cm } ^ { 2 } .$$  Find the area of the trapezium.



If   $$A B C D$$  is a parallelogram,  $$A E \perp D C$$  and  $$C F \perp A D .$$  If  $$A B = 16 \mathrm { cm }$$, $$ A E = 8 cm$$  and   $$C F = 10 \mathrm { cm } ,$$  find  $$A D .$$



In parallelogram $$ A B C D$$, $$\angle D = 115  ^ \circ ,$$  find the remaining angles .



$$ABCD$$ is a quadrilateral. $$AO$$ and $$BO$$ are the angle bisectors of angles $$A$$ and $$B$$ which meet at $$O$$. If $$\angle C=70^{o},\ \angle D=50^{o}$$, find $$\angle AOB$$.



Find the unknown angle.
1274989_36990fea2b2345dfa1da147837dd63ef.png



A photo frame is in the shape of a quadrilateral. With one diagonal longer than the other. Is it a rectangle ? why or why not ?



If the angles of a quadrilateral are in the ratio $$1:2:3:4$$  find the angles



Two angles of a quadrilateral are each of measure $$75^o$$ and the other two angles are equal. What is the measure of these two angles ? Name the possible figure so formed.



Construct a parallelogram $$ABCD$$ such that $$BC=4.5\ cm,\ BD=4\ cm$$ and $$AC=5.6\ cm$$



11 gm MNPQ, MP=3.8 cm , NQ=4.5 cm and angle between MP and NQ is $$60^{\circ}$$.



Prove that the points $$A(2, 3), B(-2, 2), C(-1, -2)$$ and $$D(3, -1)$$ are the vertices of a square $$ABCD$$.



Given a rectangle with a fixed perimeter of $$24$$ meters, if we increase the length by $$1\ \text{m}$$, then the width and area will vary accordingly.  Use the following table of values to look at how the width and area vary as the length varies.
What do you observe? Write your observations in your notebooks.
1277859_386ffca2d6e64ccca2fe5a3cdbdb1336.png



The dimensions of a rectangular room (cubiodal) are l, b and h. What is the area of its four walls?



If $$D,\ E,\ F$$ are the mid-points of the sides $$BC,\ CA$$ and $$AB$$ respectively of $$\triangle ABC$$, prove that $$BDEF$$ is a parallelogram. And also show that $$ar(\triangle DEF)=\dfrac {1}{4}ar(\triangle ABC)$$.



i\\In the figure, $$Q$$ is the centre of the circle
and $$PM$$ and $$PM$$ are tangents to the circle.
If $$\angle MPN = {40^ \circ }.$$ find $$\angle MQN.$$
1277694_e17345f9a53c4e4c9b66d80dc8b9255a.PNG



If three angles of quadrilateral  are each equal to $$75 ^ { \circ } $$ , the fourth angle is  



$$ABCD$$  is a trapezium  $$\overline { A B } \| \overline { C D }$$  If  $$A B = 20 \mathrm { cm }\mathrm { BC } = 8 \mathrm { cm } , \mathrm { CD } = 10 \mathrm { cm }$$  and  $$\mathrm { AD } = 6 \mathrm { cm } .$$ Find the area of  $$\mathrm { ABCD } .$$
1292178_b1d556e96a9f468b9ad0cfa7ef2f177a.png



In the adjoining figure  $$, ABCD$$  is a parallelogram. Any line through A cuts  $$DC$$  at a point  $$P$$  and  $$B C$$  produced at  $$Q .$$  Prove that :  $$ar ( \Delta B P C ) = ar (\Delta DPQ).$$
1293878_f8df6de59fdb466294cd8e3124285b76.1293878-Q



Diagonals  $$AC$$  and  $$BD$$  of a trapezium  $$ABCD$$  with  $$AB \parallel  DC$$  intersect each other at  $$O.$$  Prove that  $$ar ( A O D ) = a r ( B O C )$$



The angles of quadrilateral are $$x,80^{\circ},120^{\circ},100^{\circ}$$ find $$x$$



Find the value of $$x$$ in given figure.
1306895_71d44367bf814023a08d45347bf9b6af.JPG



Prove that in a parallelogram the opposite angles are equal .



PQRS is a square. Determine $$\angle SRP$$.



In the adjoining fig.  $$10.33 , A B C D$$  is a rhombus.
Find the measure of the following angles, if  $$\angle A C B = 30 ^ { \circ }$$
(a)  $$\quad \angle B O C$$
(b)  $$\angle C B O$$
(c)  $$\quad \angle O A D $$
(d)  $$\quad \angle A B O$$

1291540_c3c92c765904443f915d4bfccf567daa.png



Show that if the diagonals of a quadrilateral are equal and bisect each other at right angle, then it is a square.



Two opposite angles of a parallelogram are $$(3x-2)^{0}$$ and $$(50-x)^{0}$$. Find $$x$$



Find the area of the quadrilateral ABCD in which AB= 3 cm, BC= 4 cm,CD= 4 cm , DA=5 cm and AC= 5 cm



The bisectors of $$\angle B$$ and $$\angle C$$ of quadrilateral ABCD meet at a point $$R$$ inside the quadrilateral.If $$\angle A = 95 ^ { \circ }$$ and $$\angle D = 111 ^0,$$ find $$\angle B R C$$



The angles of a quadrilateral are in the ratio 2:3:4:Find the measure of the smallest and greatest angle.



If the mid points of the sides of a quadrilateral are joined in order, prove that the area of the parallelogram so formed will be half of the area of the given quadrilateral.



If  $$E,F,G$$  and  $$H$$  are respectively the mid-points of he sides of a parallelogram  $$ABCD,$$  show that 
$$ ar ( \mathrm { EFGH } ) = \dfrac { 1 } { 2 } ar ( \mathrm { ABCD } )$$



Diagonal AC of a parallelogram ABCD bisects $$\angle A$$
Show that:
(i) it bisects $$\angle C$$ also,
(ii) ABCD is a rhombus



Draw a quadrilateral  $$A B C D.$$  Measure the four angles  $$\angle A , \angle B , \angle C$$  and  $$\angle D.$$  Verify whether their sum is  $$360 ^ { \circ }.$$



In a quadrilateral $$ABCD$$, $$\angle A=60^{o},\angle B=105^{o}$$ and $$\angle D=50^{o}$$. Find $$\angle C$$.



Can all the four angles of a quadrilateral be obtuse (or) acute angles? For reason for your answer.



In given figure, $$P$$ and $$Q$$ are points on the sides $$AB$$ and $$AC$$ respectively of $$\triangle ABC$$ such that $$AP=3.5\ cm,PB=7\ cm,AQ=3\ cm$$ and $$QC=6\ cm$$. If $$PQ=4.5\ cm$$, find $$BC$$.

1329699_7c85d7f42b524aa9b7ad70a6d4d49316.png



A field in the form of parallelogram has one of its diagonals $$42\ m$$ and the perpendicular distance of this diagonals from either of the outlying vertices in $$10\ m\ 8\ dm$$. Find the area of the field.



A quadrilateral has three actute angles each measuring $${ 70 }^{ 0 }$$, what is the measure of the fourth angle?



$$ABCD$$ is trapezium in which $$AB \parallel DC, AB=78\ cm, CD = 52\ cm, AD=28\ cm$$ and $$BC=30\ cm$$. Find the area of the trapezium.



One side of a parallelogram is $$\dfrac{3}{4}$$ times its adjacent side. If the perimeter of the parallelogram is $$70\ cm$$, find the sides of the parallelogram .



In the given fig. $$AB||CD,\angle BDC=40^{o}$$, and $$\angle BAD=75^{o}$$. Find $$x,y,z$$.
1331202_1aab94f52b6e4713bc69befbb8a6bce4.png



Two angles of a quadrilateral measure $$80^{o}$$ each. The other two differ by $$20^{o}$$. What is the measure of each of the other angles.



In a parallelogram if the area, base & corresponding altitude are $$y^{2}, y-3, y+4$$. Then find the area of parallelogram in square unit. 



In quadrilateral $$A B C D , \angle A + \angle D = 180 ^ { \circ } .$$ What special name can be given to this quadrilateral?



Prove that the line segments joining the mid points of the opposite sides of any quadrilateral bisect each other.



In a Quadrilateral, the angles are $${x^0},{(x + 10)^0},\,{(x + 20)^0},\,{(x + 30)^{0.}}$$ Find the angles.



The ratio of two adjacent sides of a parallelogram is $$2:3$$. Its perimeter is $$50\ cm$$. Find its area if altitude corresponding to large side is $$10\ cm$$.



Prove that the points $$A(2, 3), B(-2, 2), C(-1, -2)$$ and $$D(3, -1)$$ are the vertices square $$ABCD$$.



$$ABCD$$ is a parallelogram, $$AE$$ is perpendicular to $$DC$$, If $$DC=20\ cm$$, $$AD=5\ cm$$ and the area of the parallelogram is $$40\ cm^{2}$$. Find $$DE$$.



The perimeter of a parallelogram is $$156\ cm$$. If one side measures $$36\ cm$$, find the measure of its adjacent side.



If the diagonals of a quadrilateral bisect each other then quadrilateral is a parallelograms. Prove it



Can the angles $$110 ^ { 0 } , 80 ^ { \circ } , 70 ^ { \circ }$$ and $$95 ^ { \circ }$$ be the angles of a quadrilateral? Give your reasons.



In the figure, $$\Box ABCD$$ is a parallelogram. $$AB=8, BC=12$$ and $$\angle B=30^{o}$$. Find $$A(\Box ABCD)$$.
1346363_b0b8a34740c245df91560d8c80492331.png



Show that if the diagonal of a quadrilateral interest each other at right angle, then the sum of square of opposite sides are equal;.



$$ABCD$$ is a cyclic quadrilateral. If $$\angle{BAC}={50}^{\circ}$$ and $$\angle{DBC}={60}^{\circ}$$ then find $$\angle{BCD}$$
1364461_5b351882f6c847f4bf030309de9af3dd.PNG



$$ABCD$$ is a parallelogram and $$AP$$ and $$CQ$$ are perpendiculars from vertices $$A$$ and $$C$$ on diagonal $$BD$$.Show that $$AP = CQ$$
1366210_4104ab2c0cfa4d6c9396b9db4846958e.png



What is the area of the square $$ABCD$$ shown in the diagram?
1378022_cdceae2d1f7446f2b02d8046e0f02a79.PNG



If the angles of quadrilateral are in the ratio $$3:5:9:13$$, find the value of smallest angle of quadrilateral.



Angles of a quadrilateral are $$(4x)^{\circ},\ 5(x + 2)^{\circ},\ (7x - 20)^{\circ}$$ and $$6(x + 3)^{\circ}$$. 
(i) Find the value of $$x$$ (in degrees).
(ii) Each angle of the quadrilateral 



$$ABCD$$ is a parallelogram and $$AP$$ and $$CQ$$ are perpendiculars from vertices $$A$$ and $$C$$ on diagonal $$BD$$.Show that $$\triangle{APB}\cong \triangle{CQD}$$
1366204_b3a1fb18b36e44d785ddfbfdab53b2a0.png



The parallel sides of a trapezium are 28 cm and 21 cm and each of the non-parallel sides is 17 cm. Find the area of the trapezium.



The diagonal of a rhombus are $$15$$cm and $$18$$cm. Find its area and side.



In a quadrilateral $$ABCD$$, $$ AB \parallel CD $$ . If $$ \angle A : \angle D = 2:3 $$ and $$ \angle B : \angle C = 7:8 $$, find the measure of each angle.



Prove that any line parallel to the parallel sides of a trapezium divides the non-parallel sides proportionally.



In the figure $$ABCD$$ is a rectangle and $$DEC$$ is an equilateral. Find $$\angle DAE$$.
1378248_0c5a130a27784c44893d4ba07fdda92b.PNG



The given fig is drawn using more than one triangle:
$$\angle D A B + \angle A B C + \angle B C D + \angle C D A$$
1394823_2a03e0ae4b9e4ea087e043c2b209845c.jpg



Find $$x+y+z+w$$, in the given figure.
1378390_2925477c4e634a32acb2145a132e95d4.PNG



The angles of a quadrilateral are $$(x+10), (2x+5), (2x-20)$$ & $$(2x-5)$$, then find their measures.



Find the measure of the fourth angle of the quadrilateral if the three interior angles $$48^{o},\ 85^{o}$$ and $$140^{o}$$. 



Find the four angles of equilateral, if the angles are in the ratios:
$$5:7:11:13$$



Find the angle measure $$x$$ in the following figures.
1391503_dca5b48fe9324648a73c56081132fe11.png



If $$D,E,F$$ are midpoint of the sides $$BC, CA$$ an $$AB$$ of triangle $$ABC$$, then  show that $$BDEF$$ is a parallelogram

1378277_b77a941e65944b5b872168a7d1eae60b.PNG



Find the four angles of quadrilateral, if the angles are in the ratios:
$$2:3:4:6$$



ABCD is equilateral $$\triangle BEC$$ is rectangle to BD cross ED at 'O' point $$\angle BOC=$$?
1397588_8fe68587e0824a6d8f76ddced2d03006.png



The angles in Quadrilateral are $$105,35,45,x$$ find $$x$$



In figure, $$ABCD$$ is a parallelogram 
Prove that $$\dfrac{DQ}{DP}=\dfrac{AQ}{DC}$$
1402387_09baf80d74544c49aa587c45ad50b196.png



In the figure given above, $$ABCD$$ is a cyclic quadrilateral in which $$\angle BAD=75^{\circ},\; \angle ABD=58^{\circ}$$ and $$\angle ADC=77^{\circ}.$$ Find the measure of $$\angle CBD.$$
1400722_ba339839b52e4f3b89047e6fbd60c00c.png



Do $$120,45,64,76$$ form Quadrilateral .give reasons.



Find the measure of angles x,y,z and a in the following parallelograms .
1400388_0d0567da93f641db8dc707477549cf7e.jpg



Is it possible to have a quadrilateral whose angles are of measures 105, 165, 55 and 45 . Give reason ?



From the following figure find ,
(i) x
(ii) $$\angle ABC$$
(iii) $$\angle ACD$$
1398503_fb3f28b7f9c74564a26fe298f7633016.png



$$A B C D$$ is a rhombus in which diagonals $$A C$$ and $$B D$$ intersect each other at $$O$$ . If $$\angle A C D = 55 ^ { \circ }$$ find $$\angle B D C$$



In the figure, given below, $$ABCD$$ is a cyclic quadrilateral in which $$\angle BAD=75^{o}; \angle ABD=58^{o}$$ and $$\angle ADC=77^{o}$$. Find: $$1)\angle CBD$$  $$2)\angle BCD$$
1400721_8bf91dbb27fb43debd610c393c36a31e.png



Two adjacent angles of a parallelogram are in the ratio $$1:5$$. Find all the angles of the parallelogram.



Two opposite angles of a parallelogram are $${( 3x-2)}^{\circ}$$ and $${(50-x)}^{\circ}$$.find the measure of each angle of the parallelogram.




$$ \ln a \Delta A B C, P  $$ and $$  Q $$ are points on sides $$  A B  $$ and $$  A C $$ respectively, such that PQ $$ \|\mathrm{BC}  $$ . If  $$ A P=2.4 \mathrm{cm}, A Q=2 \mathrm{cm}, Q C=3 cm$$ and $$  B C=10\mathrm{cm},  $$ find $$  A B  $$ and $$  P Q $$



Find the area of a square whose perimeter is 160 $$ \mathrm{m}  $$ . A floor 3.0 $$ \mathrm{m}  $$ long and 2.0 $$ \mathrm{m}  $$ wide is to be covered with square shaped tiles of length 10 $$ \mathrm{cm}. $$Find the cost of flooring it, if the cost of tiles is $$Rs.$$ $$150$$ per $$100$$ tiles.



In the figure, $$PQRS$$ is a parallelogram with $$PQ = 16$$ cm and $$QR = 10$$cm, $$L$$ is a point on $$PR$$ such that $$RL: LP = 2: 3$$. $$QL$$ produced meets $$RS$$ at $$M$$ and $$PS$$ produced at $$N$$.Find the lengths of $$PN$$ and $$RM$$

1406844_a53e5f6479ad449f9b649c183d47fc79.PNG



The angles of a quadrilateral are in the ratio $$3:5:9:13$$. Find all the angles of the quadrilateral.



A square $$PQRS$$ is of side $$10\, \text{cm}$$ each. Write the length of its diagonal $$PR$$.
1405416_2350446746c4451194b1fa95c13e4a5c.PNG



All the angles of a quadrilateral are equal. What special name is given to this quadrilateral?



In the given fig. $$XY\parallel BC,\,\,BE\parallel CA\,\,and\,\,FC\parallel AB.$$
Prove that $$ar\left( {\Delta ABE} \right) = ar\left( {\Delta ACF} \right).$$
1442839_a0042c4b839a4c66b20547a331f7a31c.PNG



In given figure $$ABCD$$ is a rhombus. If $$\angle BAC=38^{o}$$, find:
$$\angle DAC$$  
1456757_5c1787a898d64b2c875a7ae8ec19a67a.png




In the adjoining figure, $$ABC$$ is triangle. Through $$A,$$ $$B$$ and $$C$$ lines are drawn parallel to $$BC,$$ $$CA$$ and $$AB$$ respectively, which forms a $$\Delta PQR.$$ Show that $$2\left( {AB + BC + CA} \right) = PQ + QR + RP.$$ 
1427846_dc4f02d1882343e693425609847c6e15.png



If two sides of a cyclic quadrilateral are parallel, prove that the remaining two sides equal and the diagonals are also equal.



In given figure $$ABCD$$ is a rhombus. If $$\angle BAC=38^{o}$$, find:
$$\angle ADC$$  
1456763_d9c0ceba3d5c4f14b298e598e548ae45.png



In the quadrilateral (1) given below, $$AB||DC$$, $$E$$ and $$F$$ are mid point of $$AD$$ and $$BD$$ respectively. Prove that $$G$$ is mid point of $$BC$$.
1458348_ac597d4420c24aba98676285ed32d4d8.png



Show that opposite angles of parallelogram are equal. 



$$ABCD$$ is a quadrilateral in which $$\overline {AB}  = \overline {CD} $$ and $$\overline {AD}  = \overline {BC} $$. Show that it is a parallelogram.



In given figure $$ABCD$$ is a rhombus. If $$\angle BAC=38^{o}$$, find:
$$\angle ACB$$  
1456752_08c0f1659f9247899926a9860d7e66e4.png



Prove that opposite sides of a quadrilatral circumscribing a ciecle subtend supplementary angle at the centre of the circle.
1462271_f0386aabcb7f41cbad21c47ba939edc5.png



In the quadrilateral given below, $$AB||DC$$, $$E$$ and $$F$$ are mid point of of non-parallel sides $$AD$$ and $$BC$$ respectively. 
Calculate $$EF$$ if $$AB=6 \text{ cm}$$ and $$DC=4 \text{ cm}$$

1458398_d0fee680bfb7461d89d91d33c006e5e9.png



In the quadrilateral (1) given below, $$AB||DC$$, $$E$$ and $$F$$ are mid point of $$AD$$ and $$BD$$ respectively. Prove that
$$EG=\dfrac{1}{2}(AB+DC)$$
1458357_d2629ce4a71d47348752d2a5a2f3e758.png



In the quadrilateral given below, $$AB||DC$$, $$E$$ and $$F$$ are mid point of of non-parallel sides $$AD$$ and $$BC$$ respectively. Calculate $$AB$$, if $$DC=8 cm$$ and $$EF=9 cm$$.
1458411_abe4df55a94f479987d5d9c3defe401e.png



Show that the diagonals of a parallelogram divide it into four triangles of equal area.



$$ A B C D $$ is a quadrilateral.
Prove that $$ ( A B + B C + C D + D A ) > ( A C + B D ) $$

1458361_daa7749fb8b7450da6b407353aecfffa.png



In parallelogram $$ABCD,E$$ is the mid-point of $$AD$$ and $$F$$ is the mid-point of $$BC$$. Prove that $$BFDE$$ is a parallelogram.



If the points $$A(1,-2),B(2,3),C(a,2)$$ and $$D(-4,-3)$$ from a parallelogram. Find the value of $$'a'$$ and height of the parallelogram taking $$AB$$ as base.



Let $$ABCD$$ be a rectangle and let $$P,Q,R,S$$ be the mid-points of $$AB, BC, CD,DA$$ respectively. Prove that $$PQRS$$ is a rhombus.



The length of a diagonals of square is $$\sqrt {2}(6+2\sqrt {5})cm$$. Then finds the length of sides of the square.



The three angles of a quadrilateral are $${ 80 }^{ \circ }$$, $${ 70 }^{ \circ }$$and $${ 120 }^{ \circ }$$. The fourth angle is.



ABCD is a quadrilateral in which P, Q, R, and S are mid-points of the sides AB, BC, CD, and DA. AC is a diagonal. Show that: PQ=SR
1486499_a5f5a3d19b464052b1905b1670abe4e8.png



In parallelogram $$PQRS$$, $$\angle Q=\left( 4x-5 \right) ^{ \circ  }$$ and $$\angle S=\left( 3x-10 \right) ^{ \circ  }$$. Calculate : $$\angle Q$$ and $$\angle R$$ .



ABCD is a quadrilateral in which P, Q, R, and S are mid-points of the sides AB, BC, CD, and DA. AC is a diagonal. Show that: PQRS is a parallelogram.
1486507_90eb209d199b4203b250144351cfbf88.png



Check whether the points $$A \left(2,-1\right), B\left(3,4\right), C\left(-2,3\right)$$ and $$D\left(-3,2\right)$$ represents a Rhombus 



$$E$$ is the mid-point of the median $$AD$$ of $$\Delta ABC$$ and $$BE$$ is produced to meet $$AC$$ at $$F$$ .Show that $$AF\frac{1}{3}AC.$$



In a right angled $$\Delta BAC,\angle BAC={ 90 }^{ 0 }$$ segment AD, BE and CF are the medians
Prove that $$2\left( { AD }^{ 2 }+{ BE }^{ 2 }+{ CF }^{ 2 } \right) ={ 3BC }^{ 2 }$$



$$ABCD$$ is a quadrilateral with $$\angle{A}={80}^{\circ}, \angle{B}={40}^{\circ}, \angle{C}={140}^{\circ},\angle{D}={100}^{\circ}$$.Is $$ABCD$$ is a trapezium?Justify your answer.



In a quadrilateral , the angle are $$x^{0},(x+10)^{o},(x+20)^o,(x+30)^o$$.Find the angles



D,E and F are respectively the mid points of the sides $$AB,BC$$ and $$CA,$$ respectively of a $$\Delta ABC$$. Prove that by joining these mid-points $$D,E$$ and $$F$$ the $$\Delta ABC$$ is divided into four congruent triangle



The angles of the quadrilateral are in the ratio 3:5:9:Find all the angles of the quadrilateral.



If two angles of a quadrilateral are $$40^{\circ}$$ and $$110^{\circ}$$ and the other two are in the ratio $$3:4$$, find these angles.



The angles of a quadrilateral are $$2x,\; 2x+15,\; 4x-12$$ and $$3x+5.$$ Find the value of $$x$$ and all angles of the quadrilateral.



P,Q and R are the mid points of sides BC,CA, and AB of ABC, and AD. is . the perpendicular from A to BC. prove that P,Q,R and D are concyclic.



In figure, ABCD is a cyclic quadrilateral. If $$\angle$$BAC$$=50^o$$ and $$\angle$$DBC$$=60^o$$, then find $$\angle$$BCD.
1589001_9f2b4651f43b493ab7dd0c96dc85a254.png



The ratio between three angles of a quadrilateral is $$7:11:13$$ respectively, the value of the fourth angle of the quadrilateral is $$112^o$$. what is the difference between the largest and smallest angles of the quadrilateral ?



Show that line segment joining the midpoints of the opposite sides of a quadrilateral bisects each other.



$$ABCD$$ is a quadrilateral. Is $$AB + BC + CD + DA > AC + BD?$$
1645781_1792530b3a6646809a5b48a24a21c987.png



The diagram shows a quadrilateral on a $$1\ cm^{2}$$ grid.
Write down the mathematical name of this quadrilateral. 
1643704_1c1ef8e645f044908f9e67c41c17a8ab.png



If following statement is true enter $$1$$ else enter $$0$$.
The opposite sides of a quadrilateral are parallel. 
If one angle of the quadrilateral is $$ 60^{\circ},$$ then the other angles are $$ 120^{\circ},60^{\circ},120^{\circ} $$.



Give reasons for the following.
A rectangle can be thought of as a special parallelogram.



Find the missing values:
S.No.BaseHeightArea of the parallelogram
a.$$20\ cm$$ 
$$246$$ $$cm^2$$
b.
$$15\ cm$$ $$154.5$$ $$cm^2$$
c.
$$8.4\ cm$$ $$48.72$$ $$cm^2$$
d.$$15.6\ cm$$ 
$$16.38$$ $$cm^2$$



in a given figure a circle passes through points A,B,C and D.$$ \angle $$BAD=70$$ ^{\circ} $$,find x
1608838_056c114a6ca3461183d9057cb0d3f2af.PNG



$$ABCD$$ is quadrilateral. Is $$AB+BC+CD+DA<2(AC+BD)$$?



Give reasons for the following.
A square can be thought of as a special rectangle.



A quadrilateral has three acute angles each measure $${80}^{o}$$. What is the measure of the fourth angle?



A quadrilateral has all its four angles of the same measure. What is the measure of each?



In a quadrilateral define the following:
Adjacent sides



The angles of a quadrilateral are $${110}^{o},{72}^{o},{55}^{o}$$ and $${x}^{o}$$. Find the value of $$x$$.



Two angles of a quadrilateral are of the measure $${65}^{o}$$ and the other two angles are equal. What is the measure of each of these two angles?



The four angles of a quadrilateral are as $$3:5:7:9$$. Find the angles.



In a quadrailateral define the following:
Diagonals



Three angles of a quadrilateral are equal. The fourth angle is of measure $${150}^{o}$$. What is the measure(in degree) of equal angles?



In a quadrilateral define the following:
Adjacent angles



The three angles of a quadrilateral are respectively equal to $${110}^{o},{50}^{o}$$ and $${40}^{o}$$. Find its fourth angle. (in degrees)



The angle between the altitudes of a parallelogram, through the same vertex of an obtuse angle of the parallelogram is $$60^{\circ}$$. Find the angles of the parallelogram.



In the figure, find the measure of $$\angle MPN$$.
1672807_f038a9bde82b49739abaeb98d3ec5b5e.png



In a quadrilateral $$ABCD$$, the angles $$A,B,C,D$$ are in the ratio $$1:2:4:5$$. Find the measure of each angle of the quadrilateral.



State the midpoint theorem. 



In a quadrilateral, $$ABCD$$, $$CO$$ and $$DO$$ are the bisectors of $$\angle C$$ and $$\angle D$$ respectively. Prove that $$\angle COD=\cfrac{1}{2}(\angle A+\angle B)$$



In the figure, $$ABCD$$ is a trapezium in which $$AB\parallel DC$$. If $$\angle A={55}^{o}$$ and $$\angle B={70}^{o}$$, find $$\angle C$$ and $$\angle D$$.
1715344_295c2ca5b54e48d0bd8714c557ba7018.PNG



In the adjoining figure $$OD$$ is perpendicular to the chord $$AB$$ of a circle with centre $$O$$. If $$BC$$ is a diameter, show that $$AC\parallel DO$$ and $$AC=2\times OD$$
1715350_8758d88008ca46a2a44b9ed76c200e08.png



Three angles of a quadrilateral are $${75}^{o},{90}^{o}$$ and$${75}^{o}$$. Find the measure of the fourth angle.



In figure, AB $$||$$ CD. Find the values of x, y, z.
1677582_54ca198f4ecd47a0a69f90be8b18ab92.png



If the sum of the two angles of a quadrilateral is $${180}^{o}$$. What is the sum of the remaining two angles in degree? 



In a trapezium $$ABCD, AB || DC, AB=a\ cm$$, and $$DC=b\ cm$$. If $$M$$ and $$N$$ are the midpoint of the nonparallel sides, $$AD$$ and $$BC$$ respectively then find the ratio of $$ar(DCNM)$$ and $$ar(MNBA)$$.
1715596_33f78049493a48c982e5b5821e957cf0.png



$$P, Q, R, S$$ are respectively the midpoints of the sides $$AB, BC, CD$$ and $$DA$$ of parallelogram $$ABCD$$. Show that $$PQRS$$ is a parallelogram and also show that $$ar (||gm PQRS)=\dfrac{1}{2} \times ar(||gm ABCD)$$.
1715549_f6e114d27c5b43d7b3fce079aa2ef91a.png



In the adjoining figure, $$D$$ and $$E$$ are respectively the mid point of sides $$AB$$ and $$AC$$ of $$\triangle ABC$$. If $$PQ ||BC$$ and $$CDP$$ and $$BEQ$$ are straight lines then prove that $$ar(\triangle ABQ)=ar (\triangle ACP)$$.
1715604_474cb11880e14166965f1192552f58ed.png



In the figure, $$ABCD$$ is a parallelogram in which $$\angle A={72}^{o}$$. Calculate $$\angle B,\angle C$$ and $$\angle D$$.



Prove that the sum of all the angles of a quadrilateral is $${360}^{o}$$.
1715391_b659a5e93cd0461db7711ff755043b7a.png



In the figure, $$ABCD$$ is a parallelogram in which $$\angle DAB={80}^{o}$$ and $$\angle DBC={60}^{o}$$
1715403_4efe25abd4184949ade9e59d0d6f394f.PNG



$$P,Q,R$$ and $$S$$ are respectively the midpoints of the sides $$AB,BC,CD$$ and $$DA$$ of a quadrilateral $$ABCD$$. Show that $$PQ\parallel SR$$
1715717_f2faac8ab8b148579e4748a50cb04c7d.PNG



Prove that the line segments joining the middle points of the sides of a triangle divide it into four congruent triangles.
1715738_c41600f1470942cb9ac80f59f255cd80.png



In the figure, $$\text{D, E}$$ and $$\text{F}$$ are the midpoints of the sides $$\text{BC, CA}$$ and $$\text{AB}$$ respectively, of $$\triangle \text{ABC}$$. Show that $$\angle \text{EDF}=\angle \text{A}$$,  $$\angle \text{DEF}=\angle \text{B}$$ and $$\angle \text{DFE}=\angle \text{C}$$
1715741_397f52a2b0e245e684c0579f7e0356eb.png



$$D$$ is the midpoint of $$BC$$ of $$\triangle ABC$$ and $$E$$ is the midpoint of $$BD$$. If $$O$$ is the midpoint of $$AE$$, prove that $$ar(\triangle BOE)=\dfrac{1}{8} ar(\triangle ABC)$$.
1715527_2ae828bf55e64c70bcd99c9803f8174d.png



In the figure given below, $$ABCD$$ is a parallelogram. $$E$$ is a point on $$AB, CE$$ intersects the diagonal $$BD$$ at $$O$$ and $$EF \parallel BC$$. If $$AE : EB = 2 : 3$$, find
(i) $$EF : AD$$
(ii) area of $$\triangle BEF$$ : area of $$\triangle ABD$$
(iii) area of $$\triangle ABD$$ : area of trapezium $$AFED$$
(iv) area of $$\triangle FEO$$ : area of $$\triangle OBC$$.

1783769_3dbbdb15a6354a8f84f7c713dc177cb7.jpg



Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a square is a square.
1715746_2b8696b1d1e54d11940a0e3e9b88c009.PNG



Prove that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.
1715747_5d5c7811b57f41a8863b3303c9008cd5.png



The diagonals of a quadrilateral $$ABCE$$ are equal. Prove that the quadrilateral formed by joining the midpoints of its sides is a rhombus.
1715751_af634f619bf34522a53a187c419d90cd.png



Prove that the lines joining the middle points of opposite sides of a quadrilateral and the line joining the middle points of its diagonals meet in a point and bisect one another.



In Figure $$,\dfrac{A O}{O C}=\dfrac{B O}{O D}=\dfrac{1}{2}$$ and $$A B=5 \ {cm} .$$ Find the value of $$DC$$.


1786605_f5b67f30fd9a46a6b4c244fc5afce1ff.PNG



The diagonals of a quadrilateral $$ABCD$$ are perpendicular to each other. Prove that the quadrilateral formed by joining the midpoints of its sides is a rectangle.
1715753_0d98cd3a4ce0423da56d5ec9e06bf312.png



Fill in the blanks to make the statements true. 
The sum of all _______ of a quadrilateral is $$ 360^{\circ} $$ .



Fill in the blanks to make the statements true. 
In quadrilateral HOPE , the pairs of opposite sides are _________



Fill in the blanks to make the statements true. 
A quadrilateral in which a pair of opposite sides is parallel is ________ . 



Fill in the blanks to make the statements true. 
In quadrilateral WXYZ , the pairs of opposite angles are _________.



Fill in the blanks to make the statements true. 
If all sides of a quadrilateral are equal , it is a _______ . 



Fill in the blanks to make the statements true. 
________ is a regular quadrilateral.



Solve the following :
Two sticks each of length $$ 7 \,cm $$ are crossing each other such that they bisect each other at right angles. What shape is formed by joining their end points ? Give reason .



A line $$ l $$ is parallel to line $$m$$ and a transversal $$p$$ intersects them at $$X , Y$$ respectively . Bisectors of interior angles at $$X$$ and $$Y$$ intersect at $$P$$ and $$Q$$ . Is $$PXQY$$ a rectangle ? Give reason.



Fill in the blanks to make the statement true. 
If the diagonals of a quadrilateral bisect each other , it is a __________.



Fill in the blanks to make the statements true. 
If only one diagonal of a quadrilateral bisects the other , then the quadrilateral is known as ________



ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Then 
$$ar(\Delta BDE) = \dfrac{1}{4} ar (\Delta ABC).$$ Write True or False and justify your answer:



In the given figure, ABCD and EFGD are two parallelogram and G is the mid-point of CD. Then $$ar(\Delta DPC) = \dfrac{1}{2} ar (\parallel EFGD).$$ Write True or False and justify your answer:
1795327_d57eb038ae8d43228dd9ce1ebd67fd82.png



Find the angle measure x in the following figures:


1810208_4d52ec1db9364fc7b6b8bb889a282f7b.png



Find the angle measure x in the following figures:


1810211_583d1ee522ce40b7b8a360c43202c124.png



In trapezium $$ABCD, AB \parallel DC$$ and L is the mid-point of BC. Through L, a line $$PQ \parallel AD$$ has been drawn which meets AB is P and DC produced in Q (Fig. 9.18). Prove that $$ar (ABCD) = ar(APQD)$$
1795330_9ca6af0d40bc4b23884751292e06e0cc.png



Diagonals of a quadrilateral ABCD bisect each other. If $$\angle A = 35^\circ,$$ determine $$\angle B.$$



Can all the angles of a quadrilateral be acute angles ? Give reason for your answer.



Can all the angles of a quadrilateral be right angles ? Give reason for your answer.



Find the angle measure x in the following figures:


1810212_081504954ae04451b2b1dcb211a1777a.png



What type of quadrilateral do the points $$ A(2 , -2) , B (7 , 3) , C(11 , -1) $$ and $$ D(6 , -6) $$ taken in that order form?



Consider the following parallelograms. Find the values of $$ x, y, z $$ in each.
1810765_2f1a7e3f54824748987b04a6dd7bd183.png



Consider the following parallelograms. Find the values of $$ x, y, z $$ in each.
1810795_50dc0ba30cd1483cbd49d7bd4bf2b1c7.png



Name the quadrilateral whose diagonals are perpendicular bisectors of each other.



In the following figures, HOPE is parallelograms. Find the measures of angles $$ \mathrm{x}, \mathrm{y} $$ and $$ \mathrm{z} . $$ State the properties you use to find them.
1810845_1db6acc770e944bcb601a62d27ecfb61.png



Consider the following parallelograms. Find the values of $$ x, y, z $$ in each.
1810813_e327308a5238434e9e297dc219e12a89.png



$$ABCD$$ is a rhombus with $$P, Q$$ and $$R$$ as midpoints of $$AB, BC$$ and $$CD$$ respectively. Prove that $$PQ \bot QR$$.



In the given figure, find x+y+z+w.

1810271_9cc7b01bf56f4323b5a92e7093a9a4db.png



Name the quadrilateral whose diagonals bisect each other.



Name the quadrilateral whose diagonals are equal



Consider the following parallelograms. Find the values of $$ x, y, z $$ in each.
1810828_e65bfd277d534cb3a470600c69982c08.png



In the figure given, $$ABCD$$ is a kite in which $$BC = CD, AB =AD$$. And $$E,\ F,\ G$$ are midpoints of $$CD,\ BC$$ and $$AB$$. Prove that the line drawn through $$G$$ and parallel to $$FE$$ bisects DA.
1811126_718ed2b667da42d4a13ba110a32c03d0.png



In the adjoining figure, the lines $$l,\ m,$$ and n are parallel to each other, and G is mid-point of CD. Calculate: $$CF$$ if $$GE=2.3cm$$
1811136_69eec860936847f891f8566ccce0c214.png



In the adjoining figure, the lines $$l,\ m,$$ and n are parallel to each other, and G is mid-point of CD. Calculate:$$BG$$ if $$AD=6cm $$

1811134_84b17754a009455991df70389e1c3727.png



In the given figure, $$ABCD$$ is a parallelogram. If $$P$$ and $$Q$$ are mid-points of sides $$CD$$ and $$BC$$ respectively, show that $$CR=\dfrac14 AC$$.
1811025_8a486081c3c848f19fc0d094d6c9e36c.png



In given figure $$ABCD$$ is a quadrilateral in which $$P, Q, R$$ and $$S$$ are mid-points of the sides. $$AB, BC, CD$$ and $$DA$$. $$AC$$ is a diagonal. Show that: $$PQ = SR$$

1811076_f0b0543124fc4a56a797e313f0fa9ead.png



The diagonal $$AC$$ and $$BD$$ of a parallelogram $$ABCD$$ intersect at $$O$$. If $$P$$ is the midpoint of $$AD$$, prove that $$PO=\dfrac12\ CD$$



The diagonal $$AC$$ and $$BD$$ of a parallelogram $$ABCD$$ intersect at $$O$$. If $$P$$ is the midpoint of $$AD$$, prove that $$PQ \parallel AB$$



In the given figure, $$ABCD$$ is a parallelogram and $$E$$ is mid-points of $$AD. DL \parallel EB$$ produced at $$F$$. Prove that $$B$$ is mid-points of $$AF$$ and $$EB=LF$$.
1811022_8133db4f8f9c45b7b59b0ce62d85629a.png



In the figure given $$ABCD$$ is a parallelogram. $$E$$ and $$F$$ are mid-point of the sides $$AB$$ and $$CO$$ respectively. The straight lines $$AF$$ and $$BF$$ meet the straight lines $$ED$$ and $$EC$$ in points $$G$$ and $$H$$ respectively. Prove that $$GEHF$$ is a parallelogram.
1811093_d08793a5d3b743018795c64699e396f1.png



In given figure $$ABCD$$ is a quadrilateral in which $$P, Q, R$$ and $$S$$ are mid-points of the sides. $$AB, BC, CD$$ and $$DA$$. $$AC$$ is a diagonal. Show that: $$PQRS$$ is a parallelogram
1811077_947ef95eb40a4b5084515c8d6320893d.png



In the given figure, D, E and F are mid points of the sides BC, CA and AB respectively of $$ \Delta \mathrm{ABC} $$. Prove that BCEF is a trapezium and area of trap. BCEF $$ =3 / 4 $$ area of $$ \Delta \mathrm{ABC} $$.
1813042_b27da3579fe0462aabc9447cf2c450f3.png



In the adjoining figure, the lines $$l,\ m,$$ and n are parallel to each other, and G is mid-point of CD. Calculate: $$AB$$ if $$BC=2.4cm$$
1811140_a63235ec023049c7853f2f3e574bd7b8.png



Find the size of each lettered angle in the above figures.
1811657_9ec0baed01754575b8a97697477bfc19.JPG



In the given figure, $$ABC$$ is an isosceles triangle in which $$AB = AC$$. $$AD$$ bisects exterior angle $$PAC$$ and $$CD \, || \, BA.$$ Show that

(i) $$\angle DAC=\angle BCA $$

(ii) $$ABCD$$ is a parallelogram.



In the figure ( 2 ) given below, $$ A B C $$ is right-angled triangle at A. AGFB is a square on the side $$ A B $$ and $$ B C D E $$ is a square on the hypotenuse BC. If $$ A N \perp  ED$$, prove that:
(i) $$ \Delta \mathbf{B C F} \cong \Delta \mathbf{A B E} $$
(ii) area of square ABFG = area of rectangle BENM.
1813114_9eca97e8e2db4535bf41a070ed2a39a6.png



Prove that the quadrilateral obtained by joining the mid-points of an isosceles trapezium is a rhombus.



In the figure ( 2) given below, DE is drawn parallel to the diagonal AC of the quadrilateral ABCD to meet BC produced at the point E. Prove that area of quad. $$ \mathrm{ABCD}= $$ area of $$ \Delta \mathrm{ABE} $$
1813071_ac7a8b28ef5144d5a642060144b4d762.png



In the figure ( 2) given below, DE is drawn parallel to the diagonal AC of the quadrilateral ABCD to meet BC produced at the point E. Prove that area of quad. $$ \mathrm{ABCD}= $$ area of $$ \Delta \mathrm{ABE} $$
1813070_027ea48d98c44556a4d79b36b33b1bea.png



If E,F,G and H are the midpoint of the point of the sides AB,BC,CD and DA respectively of a parallelogram ABCD, prove that area of || gm ABCD.



In the adjoining figure, the lines $$l,\ m,$$ and n are parallel to each other, and G is mid-point of CD. Calculate: $$ED$$ if $$FD=4.4cm$$
1811141_36abc08ff966425880614691f42b646f.png



Angles of a quadrilateral are $$(4x)^o, 5(x+2)^o, (7x-20)^o$$ and $$6(x+3)^o$$. Find 
Each angle of the quadrilateral.



In quadrilateral $$ABCD$$, side $$AB$$ is parallel to side $$DC$$. If $$\angle A: \angle D =1:2$$ and $$\angle C: \angle B =4:5$$
Calculate each angle of the quadrilateral.



Use the information given in the following figure to find:
$$\angle B$$ and $$\angle C$$

1823283_1053c334e9e44da7a0d8e18f1a2d6007.png



Use the information given in the following figure to find:
$$x$$
1823279_005afb13bdd84584b288a41c81e383c4.png



From the following figure find:
$$x$$
1823301_77f682937e73409b986c7e3f74a5ff41.png



Two angles of a quadrilateral are $$89^o$$ and $$113^o$$. If the other angles are equal;
find equal angles.



One angle of a quadrilateral is $$90^o$$ and all other angles are equal; find each equal angle



If angles of quadrilateral are in the ratio $$4:5:3:6$$; find each angle of the quadrilateral.



Angles of a quadrilateral are $$(4x)^o, 5(x+2)^o, (7x-20)^o$$ and $$6(x+3)^o$$. Find 
The value of $$x$$.



Two angles of a quadrilateral are $$68^o$$ and $$76^o$$. If other two angles are in the ratio $$5:7$$, find the measure of each of them. 



In the given figure 
$$\angle b=2a+15$$
And $$\angle c=3a+5$$; find the value of $$b$$ and $$c$$
1823315_5ddc588e9af445a1b68f40edeb75f45f.png



Three angles of a quadrilateral are equal. If the fourth angle is $$69^o$$; find the measure of equal angles.
1823320_f1e143f815c74bacb90681be1a57cd0e.png



Given: In quadrilateral $$ABCD; \angle C=40^o, \angle D= \angle C$$$$  -8^o , \angle A=5(a+2)^o$$ and $$\angle B=2(2a+7)^o$$.



Each angle of a quadrilateral is $$x+5^o$$. Find
The value of $$x$$
1823381_eacb6989d74f46bca601f8eaea55c535.png



In the quadrilateral $$PQRS, \angle P: \angle Q: \angle R: \angle S=3:4:6:7$$,
Calculate to each other. Is $$PS$$ also parallel to $$QR$$?
1823328_2c9b4650ce904f37b6277588bde20e0c.png



From the following figure find:
$$\angle ABC$$
1823303_597d2b1187a447e2b950d6c68a2832d5.png



From the following figure find:
$$\angle ACD$$
1823304_4701c26879564afca2842001adaef3ea.png



The following figure shows a quadrilateral in which sides $$AB$$ and $$DC$$ are parallel. If $$\angle A:\angle D=4:5, \angle B=(3x-15)^o$$ and $$\angle C=(4x+20)^o$$, find each angle of the quadrilateral $$ABCD$$.
1823337_d691e8dcfc98478e91ca28024cd8856c.png



Two diagonals of an isosceles trapezium are $$x\ cm$$ and $$(3x-8) cm$$. Find the value of $$x$$.



Use the information given in the following figure to find the value of $$x$$.
1823333_58bd457085124fa98ae8bc17477c67d3.png



Each angle of a quadrilateral is $$x+5^o$$. Find
Give the special name of the quadrilateral taken.
1823384_73ddc2d9c8f94bf7a1ab15a9d879a65d.png



In parallelogram $$ABCD,E$$ is a mid-point of side $$AB$$ and $$CE$$ bisect angle $$BCD.$$ Prove that
Angle $$DEC$$ is a right angle.  



Each angle of a quadrilateral is $$x+5^o$$. Find
Each angle of the quadrilateral
1823383_d5d50f23c18f4331965aafa66ccc8ef6.png



In parallelogram $$ABCD,E$$ is a mid-point of side $$AB$$ and $$CE$$ bisect angle $$BCD.$$ Prove that
$$DE$$ bisects and $$\angle ADC$$ and  



In the given figure, $$M$$ is mid-point of $$AB$$ and $$DE$$, whereas $$N$$ is mid-point of $$BC$$ and $$DF$$. Show that: $$EF=AC$$.
1841072_fa4c1a23191d41c4893f5f01cb145d48.png



$$D, E$$ and $$F$$ are the mid-points of the sides $$AB, BC$$ and $$CA$$ of an isosceles triangle $$DEF$$ is also isoceles.



In parallelogram $$ABCD,E$$ is a mid-point of side $$AB$$ and $$CE$$ bisect angle $$BCD.$$ Prove that
$$AE=AD.$$



In rhombus $$ABCD:$$
If $$\angle A=74^{\circ};$$ find $$\angle B$$ and $$\angle C.$$



In parallelogram $$ABCD,\angle A=3$$ times $$\angle B.$$  Find all the angles of the parallelogram. In the same parallelogram if $$AB=5x-7$$ and $$CD=3x+1;$$ find the length of $$CD.$$



Given: Parallelogram $$ABCD$$ in which diagonals $$AC$$ and $$BD$$ intersect at $$M.$$ Prove: $$M$$ is mid-point of $$LN.$$



In trapezium $$ABCD$$, sides $$AB$$ and $$DC$$ are parallel to each other. $$E$$ is mid-point of $$AD$$ and $$F$$ is the mid-point  of $$BC$$.
Prove that: $$AB+DC=2EF$$
1841184_adfdc9a4b12d4a28b809f50536e3a589.png



In parallelogram $$ABCD, E$$ is the mid-point of $$AB$$ and $$AP$$ is parallel to $$EC$$ which meets $$DC$$ at point $$O$$ and $$BC$$ produced at $$P$$. Prove that: $$BP=2AD$$
1841173_8f3e44f3f44c4da18a81fa3cdb810939.png



In the following figure, $$AC || PS || QR || SR$$.
Prove that:
Area of quadrilateral $$PQRS=2\times $$ Area of quad $$ABCD$$.
1841224_b3423d836eb44990a3538509e85878fa.png



In each of the  following figures ,ABCD is a parallelogram in each case find the value of x and y.
1842173_25505783c7be43a5b1e130c289fc74cc.png



In the following figure, $$BD$$ is parallel to $$CA,E$$ is midpoint of $$CA$$ and $$BD=\dfrac{1}{2}CA.$$ Prove that:
$$Ar.(\triangle ABC)=2\times Ar.(\triangle DBC)$$
1841348_ac4b8c507baa4416a02d56bb3c3a7914.png



$$E,F,G$$ and $$H$$ are the midpoints of the sides of a parallelogram $$ABCD.$$ SHow that area of quadrilateral $$EFGH$$ is half of the area of parallelogram $$ABCD.$$



State 'true' or 'false'
The diagonals of a quadrilateral bisect each other.



In parallelogram  ABCD , AP and AQ are perpendicular from vertex of obtuse angle A as shown .If angles x : y = 2 : 1 ; find the angle of the parallelogram.
1842182_919ae93a982b465398f6d92f2917da92.png



In triangle $$ABC,E$$ and $$F$$ are the mid-points of sides $$AB$$ and $$AC$$ respectively. If $$BF$$ and $$CE$$ intersect each other at point $$O,$$ prove that the triangle $$OBC$$ and quadrilateral $$AEOF$$ are equal in area.



E is midpoint of sides AB and F is the midpoint of side DC pf parallelogram ABCD . Prove that AEFD is a parallelogram,



$$ABCD$$ is a parallegram. $$E,\ F$$ are the midpoints of $$BC$$ 
and $$CD$$ respectively. $$AE,\ AF$$ meet the diagonal $$BD$$ at 
$$Q$$ and $$P$$ respectively. Show that $$P$$ and $$Q$$ trisect 
$$DB.$$



Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.



$$ \square \mathrm{ABCD} $$ is a parallelogram, $$ \mathrm{P} $$ and $$ \mathrm{Q} $$ are midpoints of side $$ \mathrm{AB} $$ and $$ \mathrm{DC} $$ respectively, then prove $$ \square \mathrm{APCQ} $$ is a parallelogram.
1855916_e8c87c611ddd429f958f3578eb16e139.png



SL = LR 



In a parallelogram $$ABCD$$, If $$\angle A = (3x + 2)^o, \angle B = (2x - 32)^o$$ then find the value of $$x$$ and then find the measures of $$\angle C$$ and $$\angle D$$.



In the given figure, $$\Delta$$ABC is an equilateral triangle. Points F, D and E are the midpoints of sides AB, BC and AC respectively. Show that $$\Delta$$ FED is an equilateral triangle.
1855835_383ecbf7997e4967839ba60b210cedf5.png



In the given figure, points X, Y, Z are the midpoints of side AB, side BC and side AC of $$\Delta$$ABC respectively. AB = 5 cm, AC = 9 cm and BC = 11 cm. Find the length of XY, YZ, XZ.
1855824_e07b17c218fa44f985ea03828cf8a825.png



$$\square ABCD$$ is a parallelogram,  ratio of $$\angle A$$ to $$\angle B$$ of this parallelogram is $$5 : 4$$. Find the measure of $$\angle B$$.



LN = $$\frac{1}{2} $$ SQ



If the ratio of measures of two adjacent angles of a parallelogram is $$1:2$$, find the measures of all angles of the parallelogram.



In a quadrilateral, $$\mathrm{IJKL},$$ $$IJ\parallel KL$$, $$ \angle I=108^{\circ}, \angle \mathrm{K}=53^{\circ} \text { then find the measure of } \angle \mathrm{J} \text { and } \angle \mathrm{L}$$.



ABCD is a quadrilateral in which AD = BC and $$\angle DAB  = \angle CBA $$. Prove that
(1) $$\triangle ABD \cong \triangle BAC$$
(2) BD = AC
(3) $$\angle ABD  = \angle BAC $$



In quadrilateral ACBD, AC = AD and AB bisects $$\angle A$$. Show that $$\triangle ABC = \triangle ABD$$. What can you say about BC and BD?



In $$ \square \mathrm{ABCD}, $$ side $$ \mathrm{BC}< $$ side $$ \mathrm{AD} $$ in following figure.side $$ \mathrm{BC} \| $$ side $$ \mathrm{AD} $$ and if side $$ \mathrm{BA} \cong \operatorname{side} \mathrm{CD} $$ then prove that $$ \angle \mathrm{ABC} \cong \angle \mathrm{DCB} $$
1855968_a6b3c0927b0d4870b98910f13f5d8a11.png



In the adjacent figure, $$ \square \mathrm{ABCD} $$ is a trapezium $$ \mathrm{AB} \| \mathrm{DC} $$. Points $$ \mathrm{M} $$ and $$ \mathrm{N} $$ are midpoints of diagonal $$ \mathrm{AC} $$ and $$ \mathrm{DB} $$ respectively then prove that $$ \mathrm{MN} \| \mathrm{AB} $$.
1856109_2183400a98064244a90932245e408906.png



In the adjacent figure, seg $$ \mathrm{AB} \| $$ seg $$ \mathrm{PQ}, $$ seg $$ \mathrm{AB} \cong \operatorname{seg} \mathrm{PQ}, $$ seg $$ \mathrm{AC} \| $$ seg $$ \mathrm{PR}, $$ seg $$ \mathrm{AC} \cong \operatorname{seg} \mathrm{PR} ,$$ then prove that, seg $$ \mathrm{BC} \| $$ seg $$ \mathrm{QR} $$ and $$ \operatorname{seg} B C \cong \operatorname{seg} $$ $$QR$$.
1856089_9614da1e1d3a46759dc80d9e132c845b.png



Give a definition for each of the following terms. Are there other terms that need to be defined first ? What are they, and how might you define them ? 
Square:



In $$ \square \mathrm{ABCD}, $$ side $$ \mathrm{BC} \| $$ side $$ \mathrm{AD}, $$ side $$ \mathrm{AB} \cong $$ side $$ \mathrm{DC} $$ If $$ \angle \mathrm{A}=72^{\circ} $$ then find the measure of $$ \angle \mathrm{B}, $$ and $$ \angle \mathrm{D} $$



AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD. Show that $$\angle A> \angle C$$ and $$ \angle  B > \angle D$$.
1869076_0c469ecd49c24b5d9c1fe8cea7316674.PNG



In the given Figure, $$ \mathrm{ABCD} $$ is a trapezium. $$AB\parallel \mathrm{DC} $$. Points $$ \mathrm{P} $$ and $$ \mathrm{Q} $$ are midpoints of seg $$ \mathrm{AD} $$ and seg $$ \mathrm{BC} $$ respectively. Then prove that, $$ \mathrm{PQ} \| \mathrm{AB} $$ and $$ \mathrm{PQ}=\dfrac{1}{2}(A B+D C) $$.
1856104_8cc8be13e9a848bf8d36ab72a3ea75fa.png



$$D, E$$ and $$F$$ are respectively the midpoints of the sides $$BC, CA$$ and $$AB$$ of a $$\triangle ABC$$. Show that
$$BDEF$$ is a parallelogram.
1869551_9360130f76184cf7bd2616d28ff5f995.png



In figure $$ABC$$ and $$BDE$$ are two equilateral triangles such thta $$D$$ is the mid-point of $$BC.AE$$ intersects $$BC$$ in $$F$$. Prove that
$$ar(\triangle BDE)=\dfrac {1}{4}ar(\triangle ABC)$$
$$ar. (\triangle ABC)=2 ar.(\triangle BEC)$$
1869542_b5509ccb8fbe44eeb8b4003852e1edb3.png



Complete the hexagonal and star shaped Rangolies by filling them with as many equilateral triangles of side 1 cm as you can. Count the number of triangles in each case. Which has more triangles ? 

1869114_f80ddc5e3842433b92462f25583d0d7d.PNG



In a triangle $$ABC, E$$ is the mid-point of median $$AD$$. Show that $$ar.(\triangle BED)=\dfrac{1}{4} ar.(\triangle ABC)$$.
1869537_9dc075dc373f4cc0bf19a5c27ec985f5.png



If $$E, F, G$$ and $$H$$ are respectively the mid-points of the sides of a parallelogram $$ABCD$$, show that 
1869541_cb7d5655f68f4c59b3bba0e4a12a7a09.png



$$D, E$$ and $$F$$ are respectively the midpoints of the sides $$BC, CA$$ and $$AB$$ of a $$\triangle ABC$$. Show that
$$ar.(\triangle DEF)=\dfrac{1}{4}\ ar.(\triangle ABC)$$
1869554_c34e559c5f68431f831aa485eefb5991.png



Which of the following figures lie on the same base and between the same parallels. In such a case, write the common base and the two parallels.
1869525_cd5b57b9315a4be9b58d0eb10f29e8ca.png



In figure $$ABC$$ and $$BDE$$ are two equilateral triangles such thta $$D$$ is the mid-point of $$BC.AE$$ intersects $$BC$$ in $$F$$. Prove that
$$ar(\triangle BDE)=\dfrac {1}{4}ar(\triangle ABC)$$
$$ar. (\triangle FED)=\dfrac 18 ar.(\triangle AFC)$$
1869547_7687b0d083fd4a52a36f4c4ee3a2ff4f.png



In figure $$ABC$$ and $$BDE$$ are two equilateral triangles such thta $$D$$ is the mid-point of $$BC.AE$$ intersects $$BC$$ in $$F$$. Prove that
$$ar(\triangle BDE)=\dfrac {1}{4}ar(\triangle ABC)$$
$$ar. (\triangle BDE)=\dfrac 12 ar.(\triangle BAE)$$
1869539_70a6f9a1394f47c78d2a3c71c3c81f47.png



In figure $$ABC$$ and $$BDE$$ are two equilateral triangles such thta $$D$$ is the mid-point of $$BC.AE$$ intersects $$BC$$ in $$F$$. Prove that
$$ar(\triangle BDE)=\dfrac {1}{4}ar(\triangle ABC)$$
$$ar. (\triangle BEF)=ar.(\triangle AFD)$$
1869544_e65390a66b2c4ab885c7c70fc9e195ee.png



Complete the following:
A quadrilateral has ___ diagonals



$$P$$ and $$Q$$ are respectively the mid points of sides $$AB$$ and $$BC$$ of a triangle $$AABC$$ and $$R$$ is the mid point of $$AP$$, show that
$$ar. (\triangle RQC)=\dfrac 38 ar. (\triangle ABC)$$
1869558_003d83b509b64213b39d1eaf78aff81f.png



In fig. $$ABCDE$$ is a pentagon. A line through $$B$$ parallel to $$AC$$ meets $$DC$$ produced at $$F$$. Show that
$$ar.(\triangle ACB)=ar.(\triangle ACF)$$
1869585_cebd03b826ca4ba99e44298cfbd7be0a.png



A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decide to take over some portion of his plot from one of the corners to construct a Health Centre. Itwaari agrees to the above proposal with the condition that he should be given equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot.
Explain how this proposal with be implemented.
1869596_ca89778b2e884a25be48d89f32ef9585.png



$$BACD$$ is a trapezium with $$AB\parallel DC$$. A line parallel to $$AC$$ intersects $$AB$$ at $$X$$ and $$BC$$ at $$Y$$. Prove that area $$(ADx)=$$ area $$(ACy)$$. [Hint: Join $$Cx$$.)
1869603_98c67ac8171d45f98a03682bbe1f3258.png



In the figure ABCD is a square . M , N , O & P are the midpoint of sides AB , BC CD and DA respectively . Identify the congruent triangles.
1870987_665e6fa086d04108ba3ffe4e9923fb21.png



In fig. $$ABCDE$$ is a pentagon. A line through $$B$$ parallel to $$AC$$ meets $$DC$$ produced at $$F$$. Show that
$$ar.(\triangle EDF)=ar.(ABCDE)$$.
1869587_36c234277ced4c98aa3c91c97c481be3.png



Complete the following:
A quadrilateral has _____ sides.



$$D, E$$ and $$F$$ are respectively the midpoints of the sides $$BC, CA$$ and $$AB$$ of a $$\triangle ABC$$. Show that
$$ar.(BDEF)=\dfrac{1}{2}\ ar.(\triangle ABC)$$.
1869555_79245d7722064360b11b0c0f1fe91fe3.png



$$P$$ and $$Q$$ are respectively the mid points of sides $$AB$$ and $$BC$$ of a triangle $$AABC$$ and $$R$$ is the mid point of $$AP$$, show that
$$ar. (\triangle PRQ)=\dfrac 12 ar. (\triangle ARC)$$
1869557_b9e655f4acde4d909eb80c58cce38da0.png



In any quadrilateral $$ABCD$$, $$\angle A+\angle D={90}^{o}$$. Prove that $${AC}^{2}+{BD}^{2}={AD}^{2}+{BC}^{2}$$



Fill in the blanks:
All squares are _____.



Let $$ABCD$$ be a quadrilateral in which $$AB$$ is the smallest side and $$CD$$ is a largest side $$\angle A > \angle C$$ and $$\angle B > \angle D$$ ( Hint: join $$AC$$ and $$BD$$)



Let AB ,. CD be two segment such that $$AB | | CD $$ and $$AD  | | BC $$ . Let E be the midpoint of BC and let DE extended meet AB in F. Prove that AB = BF 



Suppose ABC is a triangle and D is the midpoint of BC . Assume that the perpendicular from D to AB and AC are of equal length . Prove that ABC is isosceles . 
1871163_1c2202e7c7a94ad681de8c73a165b3b2.png



Let ABCD be a quadrilateral in which AD is the largest side and BC is the smallest side. Prove that $$ \angle  A < \angle C$$ [Hint: join AC ] 
1871198_c0b8a24abd1c4d6b83a6f262649379b4.png



In a triangle $$ABC$$, let $$D$$ be the midpoint of $$BC$$. Prove that $$AB+AC > 2AD$$ (what property of quadrilateral is needed here )



Let $$ABCD$$ be a quadrilateral and $$P, Q, R, S$$ be the midpoints of $$AB, BC, CD, DA$$ respectively prove that $$PQRS$$ is a parallelogram [ what extra result you need to prove this ] It can be proved if $$ABCD$$ is a rectangle



Prove that midpoint (C) of hypotaneous in a right-angled triangle AOB is situated at equal distance from vertices $$ 0, \mathrm{A} $$, and $$ \mathrm{B} $$ of triangle.



Let $$ABCD$$ be a quadrilateral with diagonals $$AC$$ and $$BD$$. Prove the following statement
$$AB+BC+CD > AD$$



Prove that the line segment joining the mid-point of the hypotenuse of a right triangle to the vertex of the right angle is equal to half the hypotenuse.



Show that the line segment joining the mid-points of the consecutive sides of a square is also a square.




In figure, $$ABCD$$ is a parallelogram $$P$$ and $$Q$$ are the mid-points of opposite sides $$AB$$ and $$DC$$ of a parallelogram $$ABCD$$. Prove that $$PRQS$$ is a parallelogram.
1878790_6bcd402725824c248847bd6a37f6e219.png



In the given figure, $$ABCD$$ is a trapezium in which $$AB \parallel CD$$. $$M$$ and $$N$$ are mid-points of the diagonal $$AC$$ and $$BD$$ respectively. Prove that $$MN \parallel CD$$ and $$MN= \dfrac12 (CD- AB)$$.
1879353_94d97056bf2f4e278de8116a42f1f21b.png



One angle of a quadrilateral is $$108^\circ$$ and the other three angles are equal. Find the value of each of the equal angle.



In a quadrilateral $$ABCD, \angle A = 3x, \angle B= 5x, \angle C= 20x$$ and $$\angle D= 8 x$$. Find the value of $$x$$.



Can all the four angles a quadrilateral are obtuse? Give reason for your answer.



The three angles of a quadrilateral measure $$56^\circ, 100^\circ$$ and $$88^\circ$$. Find the measure of the fourth angle.



In figure, the diagonal $$AC$$ of a quadrilateral $$ABCD$$ bisects the angle $$A$$ and $$C$$. Prove that $$AB = AD$$ and $$CB = CD$$.

1878341_b4a2042946a14c848b50de8e99a91a43.png



State whether the following statement is true or false, give a reason. The angles of a certain quadrilateral are $$ 50^{\circ}, 60^{\circ}, 112^{\circ}, 130^{\circ} $$



Let $$ABCD$$ be a unit square. Draw a quadrant of a circle with $$A$$ as the center and $$B,\ D$$ as endpoints of the arc. Similarly, draw a quadrant of a circle with $$B$$ as the center and $$A,\ C$$ as endpoints of the arc. Inscribe a circle $$\Gamma$$ touching the arc $$AC$$ externally, the arc $$BD$$ internally, and also touching the side $$AD.$$ Find the radius of the circle $$\Gamma$$.



Suppose $$EFGH$$ is a trapezium in which $$EF$$ is parallel to $$HG$$. Through $$X$$, the mid-point of $$EH$$, $$XY$$ is drawn parallel to $$EF$$ meeting $$FG$$ at $$Y$$. Prove that $$XY$$ bisects $$FG$$.



Suppose $$X$$, $$Y$$ and $$Z$$ are the midpoints of the sides $$PQ$$, $$QR$$ and $$RP$$ respectively of a triangle $$PQR$$. Prove that $$XYRZ$$ is a parallelogram.



The base of a pyramid is a rhombus whose diagonals are 6 m and 8 m long. The altitude of the pyramid passes through the point of intersection, of the diagonals of the rhombus and is 1 m long. Find the lateral area of the pyramid.



Prove that if the mid-points of the opposite sides of a quadrilateral are joined, they bisect each other.



In a rectangle $$ABCD$$, $$P$$, $$Q$$, $$R$$ and $$S$$ are the midpoints of the sides $$AB$$, $$BC$$, $$CD$$ and $$DA$$ respectively. Find the area of $$PQRS$$ in terms of area of $$ABCD$$.



Prove the converse of the mid-point theorem following the guidelines given below: Consider a triangle $$ABC$$ with $$D$$ as the mid-point of $$AB$$. Draw $$DE \parallel BC$$ to intersect $$AC$$ in $$E$$. Let $${E}_{1}$$ be the mid-point of $$AC$$. Use mid-point theorem to get $$D{E}_{1} \parallel BC$$ and $$D{E}_{1} = {BC}/{2}$$. Conclude $$E = {E}_{1}$$ and hence $$E$$ is the mid-point of $$AC$$.



Prove that the sum of four angles of a quadrilateral ABCD is $$360^{0}$$.



Ina quadrilateral $$ABCD$$, $$AB=AD$$ and $$CB=CD$$. Prove that :(i) $$AC$$ bisects angle $$BAD$$.(ii) $$AC$$ is perpendicular bisector of $$BD$$.



In quadrilateral ABCD $$\angle$$B$$=90^o$$, $$\angle$$C $$-\angle$$D$$=60^o$$ and $$\angle$$A $$-\angle$$C$$-\angle$$D $$=10^o$$. Find $$\angle$$A, $$\angle$$C and $$\angle$$D.
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In a parallelogram $$ABCD,$$ $$\angle A = x ^ { \circ } , \angle B = ( 3 x + 20 ) ^ { \circ }$$ . Find $$x$$ and $$\angle C$$ and $$\angle D$$



$$ABCD$$ is a parallelogram and $$\angle DAB=60^{o}$$. If the bisects $$AP$$ and $$BP$$ of angles $$A$$ and $$B$$ respectively, meet at $$P$$ on $$CD$$.Prove that $$P$$ is the midpoint of $$CD$$.



In the adjacent figure, $$ar(PEA)=ar(PAC)$$ and $$ar(RPC)=ar(KEA)$$ show that quadrilateral $$PARK$$ and $$PACE$$ are trapezium.
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$$ABCD$$ is a rhombus and $$P, Q, R$$ and $$S$$ are the mid-points of the sides $$AB, BC, CD$$ and $$DA$$ respectively. Show that the quadrilateral $$PQRS$$ is a rectangle.



In the given figure, ABCD is a trapezium, Diagonals AC and BD interset at Q Prove that the area of $$\Delta $$AOD is the same as  $$\Delta $$BOC



$$ABCD$$ is a rhombus. If $$\angle BCA=35^{o}$$, find $$\angle ADC$$.



State the angle sum property of a quadrilateral.



The area of a rectangular plot is 540 $${ m }^{ 2 }$$. If its length is 27 m, find its breadth and perimeter.



$$ABCD$$ is a parallelogram. The circle through $$A, B$$ and $$C$$ intersects $$CD$$ produced at $$E$$. If $$AB=10\ cm, BC=8\ cm, CE=14\ cm$$ . Find $$AE$$



Parallelogram $$ABCD$$ and rectangle $$ABEF$$ are on the same base and have equal areas. Show that perimeter of the parallelogram is greater than that of rectangle.
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In the figure, $$\Box ABCD$$ is a trapezium, $$AB\parallel DC$$. Point $$P$$ and $$Q$$ are midpoints of $$seg\ AD$$ and $$seg\ BC$$ respectively. Then prove that, $$PQ\parallel AB$$ and $$PQ=\dfrac{1}{2}(AB+DC)$$.
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If the tangents  $$P A$$  and  $$P B$$  from a point  $$P$$  to the circle with cetre  $$O$$  inclined to each other at the angle of  $${ 110 }^{ },$$  then find  $$\angle POA.$$



In a parallelogram, the bisector of any two consecutive angles intersect at right angle, Prove it.



The diagonals of a rectangle ABCD intersect at o. if $$\angle $$ BOC =$$70^{\circ}$$. Find $$\angle $$ODA.



PQRS is a parallelogram. PM is the height from P to $$\overline {SR} $$ and PN is the height from P to $$\overline {QR} $$. If SR=12 cm and PM=7.6 cm. Find the area of the parallelogram PQRS.
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Both the pair of opposite angles of quadrilateral are equal and supplementary. Find the measure of each angle. 



If non parallel sides of a trapezium are equal, prove that it is cyclic.



Line segment joining the mid - points M and N of parallel sides AB and DC respectively of a trapezium ABCD is perpendicular to both the sides AB and DC . Prove that AD = BC.



In triangle ABC, P is the mid-point of side BC. A  line through P and parallel to CA meets AB at point Q; and a line through Q meets at $$S$$. QS parallel to BC meets median AP at point R. prove that : (i) AP = 2AR (ii) BC= 4QR



In the above figure RISK and CLUE are parallelograms, Find the value of x.
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In figure above, the bisectors of $$\angle P$$ and $$ \angle Q$$ meet at a point O. If $$\angle R={ 100 }^{ \circ }$$ and $$\angle S={ 50 }^{ \circ}$$, find $$\angle {POQ}$$.
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Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementarily angles at the center of the circle.



In the adjoining figure, D, E and F are mid points of the side BC, CA and AB of $$\Delta ABC$$. If $$AB =4.3$$ cm , $$BC =5.6$$ cm and $$AC =3.5$$ cm. Find the perimeter of 
(i) $$\Delta DEF$$   (ii) Quad. BDEF
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$$ABCD$$ is a trapezium in which $$AB || DC, AB=16\ cm$$ and $$DC=24\ cm$$. If $$E$$ and $$F$$ are respectively the midpoints of $$AD$$ and $$BC$$, prove that $$ar(ABEF)=9/11\ ar (EFCD)$$.



Image 
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The length of the diagonals of a rhombus is in  the ratio 4:3 .If its area is 384 $$cm^{2}$$, find its side.



Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rhombus is a rectangle.
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Bisectors of $$\angle{B}$$ and $$\angle{D}$$ of a quadrilateral ABCD meet CD and AB produced at P and Q respectively. Prove that $$\angle{P} + \angle{Q} = 1/2(\angle{ABC} + \angle{ADC})$$



$$P,Q,R$$ and $$S$$ are respectively the midpoints of the sides $$AB,BC,CD$$ and $$DA$$ of a quadrilateral $$ABCD$$, Show that $$PQ\parallel AC$$ and $$PQ=\cfrac{1}{2}AC$$
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$$ABCD$$ is a trapezium in which $$AB \parallel DC$$ and its diagonals intersect each other at $$O$$. Using Basic Proportionality theorem, prove that $$AO/BO = CO/DO$$.



The midpoints of the sides $$AB,BC,CD$$ and $$DA$$ of a quadrilateral $$ABCD$$ are joined to form a quadrilateral. If $$AC=BD$$ and $$AC\bot BD$$ then prove that the quadrilateral formed is a square.
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If $$D,\ E$$ and $$F$$ mid-points of sides $$AB,\ BC$$ and $$CA$$ respectively of an isosceles triangle $$ABC$$, prove that $$\triangle DEF$$ is also isosceles.



The angles of quadrilateral are  in the ratio 3 : 4 : 5 : 6 Show that the quadrilateral is a trapezium.



Class 9 Maths Extra Questions