Basic Geometrical Ideas - Class 6 Maths - Extra Questions

Define the following terms: Intersecting lines

Lines that have one and only one point in common are known as intersecting lines.
A minimum of two lines are required for intersection
The common point where all the intersecting lines meet is called the Point of Intersection


Explain how a square is:
(i) a quadrilateral (ii) a parallelogram (iii) a rhombus (iv) a rectangle

(a) A quadrilateral has four sides
(b) A parallelogram's opposite sides are parallel.
(c) A rhombus's four sides are of same length
(d) A rectangle's interior angles are equal to $$90^{\circ}$$.

All of these properties are followed by Square.


Identify all the quadrilaterals that have
(a) four sides of equal length 
(b) four right angles

In geometry, a quadrilateral can be defined as a closed, two-dimensional shape that has four straight sides. 

Fig A shows Rhombus, B shows a rectangle, C shows a square.  

(a) Quadrilaterals that have four sides of equal length are Rhombus and squares.
(b) Quadrilaterals that have four right angles are Rectangle and squares.
489296_463386_ans_992c3440f67441ddba8b9525ab6ec99a.png


Find the value of x in the following figure.
570655.jpg

$$\angle AOB+\angle BOC={ 90 }^{ \circ  }\\ { 40 }^{ \circ  }+{ (6x+2) }^{ \circ  }=90^{ \circ  }\\ 6x+{ 42 }^{ \circ  }=90^{ \circ  }\\ 6x=90^{ \circ  }-42^{ \circ  }=48^{ \circ  }\\ x=\dfrac { 48^{ \circ  } }{ 6 } =8^{ \circ  }$$


Identify the positions of the given $$6$$ points with respect to the angle. How many of the shown points lie in the interior of the angle?
636547_2ac40c64649c444ea7230e95fcd1ec16.png

Any point lying between the arms of an angle is interior point of angle. Here three points are lying between the arms.
So answer is $$3$$.


How many vertices does an angle have?

The point where the arms of the angle meet is called vertex of angle. An angle has one vertex.
In the figure $$B$$ is the vertex of $$\angle ABC$$
So correct answer is $$1$$

836952_636496_ans_1de3ce21334b4c4c8b214be8fba6d4ea.jpg


How many arms does an angle have?

Two lines are required to form an angle. Therefore, an angle has two arms.
So the correct answer is $$2$$.

836949_636495_ans_76df93c89ae7482f95e03472c48b74a8.jpg


Identify the positions of the given $$6$$ points with respect to the angle. How many of the shown points lie in the exterior of the angle?
636547_2ac40c64649c444ea7230e95fcd1ec16.png

Any point lying outside the arms of an angle is exterior point of angle . Here two points are lying outside the arms.
So answer is $$$2$$.


Prove  that the sum of the three angles of a triangle is two right angles. If in a right angles triangle an acute angle is 1/4th the other, find the acute angles.

As we know that the sum of all the three angles of a triangle is $$180^\circ$$ and a right angle is $$90^\circ$$.
Number of right angles in a triangle $$=\dfrac{180^\circ} {90^\circ}=2$$
Hence proved.

Say $$X =$$ smaller acute angle in right triangle 
$$4X =$$ larger acute angle 
$$90^\circ =$$ right angle 

$$x + 4x + 90^\circ = 180$$ 
$$5x = 90^\circ$$ 
$$x = 18^\circ$$ 
$$4x = 72^\circ$$

Acute angles are $$18^\circ$$ and $$72^\circ$$


If one of the angles of triangle is $${50^ \circ }$$ and the other two angle are equal, find the measure of each of equal angles

Consider the given question,
$$\theta _1=50^{\circ},$$ and other two angles are same, $$\theta _2=\theta _3$$

We know that in a triangle,
$$ \theta_1+\theta _2+\theta _3=180^{\circ}$$

$$\Rightarrow 50^{\circ}+\theta _2+\theta _2=180^{\circ} $$
$$\Rightarrow \theta _2=\dfrac{130}{2}^{\circ} $$
$$\Rightarrow \theta _2=65^{\circ} $$

Because $$\theta _2=\theta _3$$
$$\theta _3=65^{\circ}$$

Hence, the value of angle is $$\theta_2=\theta _3=65^{\circ}$$


In the figure $$ABCD , BD= BC = AD $$ and $$\angle ACD = {37^ \circ }$$. find $$\angle ADB$$
1065797_21ef28998d6b494cac74fca1b870e1b4.png

Since, $$BD=BC=AD$$
Hence, $$\angle BDC=\angle BCD=\angle ACD=37^\circ$$
$$\angle ACD=\angle ABD=37^\circ$$                                        (Opposite angles of $$AD$$)
$$\angle ADB=y$$

In $$\triangle BCD,$$
$$\angle BCD+\angle CDB+\angle DBC=180^\circ$$
$$37^\circ+37^\circ+\angle DBC=180^\circ$$
$$\angle DBC=106^\circ$$
Now,
$$\angle DBC=106^\circ=\angle DAC=\angle BAD$$                 (Opposite angles of side $$CD$$)        

In $$\triangle ABD$$,
$$AD=BD$$
$$\angle ABD=\angle BAD=37^\circ$$
$$\angle BAD+\angle ADB+\angle DBA=180^\circ$$
$$106^\circ+37^\circ+y=180^\circ$$
$$y=37^\circ$$
This is the required solution.


What are distinct intersecting lines?

Distinct intersecting lines are lines(usually two lines) that intersect at one and only one point.
In the above image, two lines $$l, \ m$$ intersect at point $$K$$.
Therefore, Lines $$l, \ m$$ are two distinct lines

1007594_1057976_ans_b551f2a2ebcd4862930d0094a7ce9d35.JPG


Prove that the two lines that are respectively perpendicular to two intersecting lines intersect each other.


If $$n$$ and $$p$$ are any two intersecting lines.

Draw two lines such that $$l\perp n$$ and$$m\perp p$$


To prove: Lines $$l$$ and$$m$$ will intersect.


Proof:

Let $$n$$ and $$p$$ be two intersecting lines.

Let us assume that $$l$$ and $$m$$ do not intersect each other.

Now $$l\perp n$$              (Given)

Let us assume that $$l\perp m$$

Therefore, $$m\perp n$$              $$...(1)$$

But given that $$m\perp p$$         $$...(2)$$

From $$(1)$$ and $$(2),$$ we get $$n\parallel p$$ 

But given, lines $$n$$ and $$p$$ intersect each other. 


Hence, our assumption is wrong.

Therefore $$l$$ and $$m$$ intersect each other.
1053009_1083602_ans_11793f68e0dc449d983a69fe8edb3bdd.png


Two angles form a linear pair, the measure of one angle is twice the measure of the other angle. Find the measure of both these angles.

If two angles form a linear pair, then the sum of the degree measure of the angles is $$180^o$$.
Let $$x$$ be the measure of the smaller angle.
Then,
$$2x$$ is the measure of the other angle.
So,
$$x+2x=180^o$$
$$3x=180^o$$
$$x=60^o$$
The measures of the angles are $$60^o$$ and $$120^o$$.


The product of two consecutive positive integers isFormulate the quadratic equation whose roots are these integers.

Let two consecutive positive integers be $$x$$ and $$x+1$$. Then, their product is $$x(x+1)$$.
It is given that product is 240.
Therefore,
$$x(x+1)=240$$
$$x^{2}+x-240=0$$
This is the required quadratic equation.


How many lines can pass through two distinct points.

$$only\ one\ line$$


Find the value of $$x$$ and $$y$$.
1338296_98abe64db1dd4c61982136482080d88e.png

From the fig. given-
$$\left( x + 40 \right) + 3x = 180$$
$$\Rightarrow 4x = 180 - 40$$
$$\Rightarrow x = \cfrac{140}{4} = 35$$
Also,
$$3x + y = 180$$
$$\Rightarrow y = 180 - 3 \times 35$$
$$\Rightarrow y = 75$$


Find the 'x'
1204615_788c8c7d07294dbe82958fff83e00927.PNG

$$\angle x=30°$$


What is polygon ?

Any figure in a plane bounded by three or more line segments is called a polygon.


The supplement of an angle is one-third of itself. Determine the angle and its supplement.

Let the measure of the angle be $$x^\circ$$. Then, the measure of its supplementary angle is $$(180-x)^\circ$$.

According to given information,

$$180-x=\dfrac{1}{3}x$$

$$3(180-x)=x$$

$$4x=540$$

$$x=135^\circ$$

Supplementary angle $$=180^\circ-135^\circ=45^\circ$$

Thus the measure of the angle is $$135^\circ$$ and the measure of its supplement is $$45^\circ$$.


Fill in the blanks:
A four sided polygon is called a ____

A four sided polygon is called a Quadrilateral.
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Define the following term : Intersecting lines 


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Name the vertex and the arms of $$\angle ABC$$ given in the figure.

1396619_5ec8494b25a94764820cafb56bd5c072.png

Arms of $$\angle ABC$$ are rays $$AB$$ and $$AC$$ and its vertex (intersection of arms) is $$B.$$


Use the figure to name two pairs of intersecting lines.
1644291_db9746f1619f493e816d58288a31fb1e.png

Step 1: To write
(1) Name of two pairs of intersecting lines

Step 2: Pairs of intersecting lines
Intersecting lines have one common point at the intersection.
Therefore, two pairs are intersecting lines are:
(1) Line $$\overleftrightarrow{OC}$$ and line $$\overleftrightarrow{AE}$$ 
(2) Line $$\overleftrightarrow{AE}$$ and line $$\overleftrightarrow{EF}$$


Draw any line segment $$\overline{AB}$$. Mark any point M on it. Through M, draw a perpendicular to $$\overline{AB}$$. (Use ruler and compasses)

$$(1)$$ Draw the given line segment $$\overline{AB}$$ and mark any point M on it.
$$(2)$$ With M as centre and a convenient radius, construct radius, construct an arc intersecting the line segment $$\overline{AB}$$ at two points C and D.
$$(3)$$. With C and D as centres and a radius greater than CM, construct two arcs. Let these be intersecting each at E.
$$(4)$$ Join EM. $$\overline{EM}$$ is perpendicular to $$\overline{AB}$$.
1336409_1645095_ans_d7fa91e820684f1bb3c4fc64a73e22fc.png


Draw rough diagram to illustrate the following.
Closed curve.

Step 1: To draw
(1) A closed curve

Step 2: Rough Sketch of a closed curve





Examine whether the following is a polygon or not. If not, say why?
1644535_d57b069199a74cbf9590d43337cdc2bb.png

Since the figure is not made up of line segments, so it is not a polygon.


In how many points can two distinct lines at the most intersect ? 

Suppose we have two lines $$L_1$$ and $$L_2$$.
If $$L_1$$ and $$L_2$$ are parallel, then they do not intersect.
If $$L_1$$ and $$L_2$$ are not parallel, then they will intersect at a single point.
Hence two distinct lines can intersect at most at one point.

1522611_1669218_ans_013437e1906e4343b6de2173d570be7f.png


From the given figure, write line whose point of intersection is $$E$$.
1787302_3e95b934e1f14c17b864fc4f9a47e1ca.png

As we can see in the diagram, the lines whose point of intersection is $$E$$ are : $$l$$ and  $$p$$.
These both are called Intersecting Lines.


If one of four angles formed by two intersecting lines is a right angle, then show that each of the four angles is a right angle.


1653492_1669258_ans_0c1caa39d4844ba9b9d3b13927bc3561.jpg


In the given figure. Name concurrent lines.
1787323_fde46006c72548eb85dd1c6905b224dc.png

In the figure, concurrent lines are : $$\overline{AD}, \overline {ED}, \overline  {CD}$$


Use the figure to name
Line passing through A.
1787761_df2272218ed3416eb2a8554549814318.png

$$\overleftrightarrow { AE }  $$, etc


In Fig., name the following :
Two pairs of non-intersecting line segments.
1669257_014dc6bb2aeb43178cd134293676a39d.png


1653488_1669257_ans_ba966aa714a2405988162b38a116f781.jpg


Define the following term : 
Interesting Lines

Two lines having common point are called intersecting lines.


Fill in the blanks in each of the following to make the statement true:
If two parallel lines are intersected by a transversal, then each pair of corresponding angles are _______


1656333_1669378_ans_2b2b6977879c4c44b1ce588e1dc02379.jpg


How many lines can be drawn through two distinct given point?

One
1719631_1787730_ans_18af3be99d104b3e995ea5d202d78b90.PNG


Use the figure to name:
A line
1787747_290a3141f88d4d56bcf8922738d099e4.png

$$\overleftrightarrow { DE } ,\overleftrightarrow { DO } ,\overleftrightarrow { DB } ,\overleftrightarrow { EO } $$, etc


Fill in the blanks:
There is exactly one line passing through _____ distinct points in a plane.

There is exactly one line passing through $$2$$ distinct points in a plane.


Fill in the blank:
A curve which does not cross itself at any point is called a ....... curve.

A curve which does not cross itself at any point is called a simple curve.


Fill in the blank:
A simple closed curve made up entirely of line segments is called a _____.

A simple closed curve made up entirely of line segments is called a polygon.


In figure, how many points are marked? Name them.
1789163_c3266bf810124441bac22aea654eedcc.png

In the given figure, A, B, C, D and E are the only points which are marked.
$$\textbf{Hence, five points are marked in the given figure.}$$ 



How many line segments are there in figure?
1789149_8f256f3bc28c4e19b8a075e04109dd94.png

$$\textbf{Finding the number of line segments.}$$ 
            Shortest distance between two given points is known as the line segment.
           As there are only two points in the given figure namely A and B.
           Hence, from two points only 1 line can pass.
$$\textbf{Hence, AB is the only one line segment in the given figure.}$$



The distance around a circle is its _____ .

The distance around a circle is its Circumference.
$$C=2\pi r$$



State the midpoints of all the sides of figure.
1788745_b33dc1a52a4c404a8aea24eebfa073ab.png

$$Y$$, midpoint of $$CB$$
$$X$$ , midpoint of $$AC$$
$$Z$$, midpoint of $$AB$$


Can you identify the regular quadrilateral?

Square is regular quadrilateral with all it's sides equal.


How many lines can pass through 
(a) one given point ?
(b) two given points ?

a) infinite number of lines can pass through a single point.
b) Only one line can pass through two given points


State whether True or False.
(a) All rectangles are squares.
(b) All rhombuses are parallelograms.
(c) All squares are rhombuses and also rectangles.
(d) All squares are not parallelograms.
(e) All kites are rhombuses.
(f) All rhombuses are kites.
(g) All parallelograms are trapeziums.
(h) All squares are trapeziums.

(a) False
All squares are rectangles. 
(All sides of rectangle are not equal).

(b) True
Opposite sides are equal and parallel.

(c) True
All sides of the rhombus are equal 
$$\therefore$$ All squares are rhombuses. 
All squares are also rectangles as each interval angle measure $$90^{\circ}$$

(d) False
All squares are parallelogram as opposite sides are equal and parallel.

(e) False
Kite has different lengths.

(f) True
Rhombus also has two distinct consecutive pairs of sides of equal length.

(g) True
Parallelogram pair of parallel sides.

(h) True
All squares have a pair of parallel sides.


Name the quadrilaterals whose diagonals.
(i) bisect each other 
(ii) are perpendicular bisectors of each other 
(iii) are equal

(i) Bisects each other: Diagonals of a parallelogram, rhombus, square and rectangle.
(ii) Are perpendicular bisectors of each other: Diagonals of rhombus and square $$\rightarrow$$ perpendicular bisectors.
(iii) Are equal: Diagonals of rectangle and square are equal.


Given here are some figures:
Classify each of them on the basis of the following
(a) Simple curve 
(b) Simple closed curve 
(c) Polygon 
(d) Convex polygon 
(e) Concave polygon

463354.png

A curve is said to be simple if it does not cross itself.
(a) Simple curve $$\rightarrow 1, 2, 5, 6, 7$$

A simple closed curve is that curve which forms a path which starts and ends at the same point. 
(b) Simple closed curve $$\rightarrow 1, 2, 5, 6, 7$$

A polygon is a two dimensional closed figure which consists of sides and vertices.
(c) Polygon $$\rightarrow 1, 2$$

A polygon in which the measure of each interior angle is less than $$180^o$$ is called as a convex polygon.
(d) Convex polygon $$\rightarrow 2$$

A polygon in which the measure of one interior angle is greater than $$180^o$$ is called as a concave polygon.
(e) Concave polygon $$\rightarrow 1$$


How many diagonals does each of the following have?
(a) A convex quadrilateral 
(b) A regular hexagon 
(c) A triangle


(a) There are $$2$$ diagonals in a convex quadrilateral.
(b) There are $$9$$ diagonals in a regular hexagon.
(c) A triangle does not have any diagonal in it.
489188_463355_ans.png


What is a regular polygon?
State the name of a regular polygon of
(i) $$3$$ sides (ii) $$4$$ sides (iii) $$6$$ sides

Regular Polygon: A polygon with all sides and angles equal.
(i) $$3$$ sides: Equilateral triangle 
(ii) $$4$$ sides: Square 
(iii) $$6$$ sides: Regular Hexagon


$$ABC$$ is a triangle right angled at $$C$$. A line through the midpoint $$M$$ of hypotenuse $$AB$$ and Parallel to $$BC$$ intersects $$AC$$ and $$D$$. Show that $$MD\perp AC$$.
699539_8c914cc236174a4d8d8dfacb590c5fa7.png


1150312_699539_ans_5663782d764c439b83b10b912fefa713.png


$$PQ,QR$$ and $$RS$$ are three consecutive sides of a regular polygon. If <$$QPR=20^o$$ find the number of side in the polygon.

angle QPR = 20

So angle QRP = 20 (QR=QP)


angle RQP = 180-40(angle sum property of a triangle)

                     =140 
So one  exterior angle = 180 -140 = 40

$$\therefore$$ number of sides = $$\dfrac{360}{40} = 9$$ sides


An angle is greater thanIs its complementary angle greater than 45 or equal to 45 or less than 45?

let $$\angle a \ and \angle b$$ be the two complementary angles.
so,
$$\angle a + \angle b=90$$

Here, one angle is 45

Let $$\angle a=45$$

$$\Rightarrow 45+\angle b=90$$

$$\angle b=90-45=45$$

So from above we can say that if one complementary angle is greater than 45 then other angle is less than 45.





Find the ratio in which the line joining $$(-2, 5)$$ and $$(-5, -6)$$ is divided by the line $$y=-3$$. Hence find the point of intersection.

$$\rightarrow$$ Let $$x_{1}, y_{1}=(-2, 5)$$
$$x_{2}, y_{2}=(-5, -6)$$
And here $$y=(-3)$$
$$\therefore y=\dfrac{m_{1}y_{2}+m_{2}y_{1}}{m-[1]+m_{2}}$$
$$\therefore (-3)=\dfrac{m_{1}(-6)+m_{2}(5)}{m_{1}+m_{2}}\rightarrow (1)$$
$$\therefore -3m_{1}-3m_{2}=-6m_{1}+5m_{2}$$
$$\therefore -3m_{1}+6m_{1}=5m_{2}+3m_{2}$$
$$\therefore 3m_{1}=8m_{2}$$
$$\therefore\boxed{\dfrac{m_{1}}{m_{2}}=\dfrac{8}{3}}$$
$$\rightarrow$$ The ratio will be $$m_{1}: m_{2}=8:3$$
$$\rightarrow$$ To find equation of line passing through the above co-ordinates we use intercept forms:
$$y-y_{1}=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})$$
$$\therefore y-5=\dfrac{-6-5}{-5(-2)}(x-(-2))$$
$$\therefore y-5=\dfrac{-11}{-3}(x+2)$$
$$\therefore y-5=\dfrac{11}{3}(x+2)$$
$$\therefore 3y-15=11x+22$$
$$\therefore 3y=11x+37$$
$$\therefore \boxed{y=\dfrac{11}{3}x+\dfrac{37}{3}}\Rightarrow$$ equation of another line
$$\rightarrow$$ To find intersection of both the line
$$11x+37=3y$$
$$\therefore 11x-3y+37=0 \rightarrow (1)$$
$$y+3=0 \rightarrow (2)$$
$$\therefore$$ From eq $$(2)$$ we get
$$\boxed{y=(-3)}$$
$$\rightarrow$$ Putting value of $$y$$ in eq $$(1)$$
$$11x-3(-3)+37=0$$
$$\therefore 11x+9+37=0$$
$$\therefore \boxed{x=\left(\dfrac{-46}{11}\right)}$$
Therefore point of intersection
$$\left(\dfrac{-46}{11}, -3\right)$$


Fill in the blanks using the correct word given in brackets:
Two polygons of the same number of sides are similar, if (a) their corresponding angles are and (b) their corresponding sides are ..........(equal, proportional).

Two polygons having same number of sides are similar, if there corresponding angles are equal and their corresponding sides are proportional

For polygons to be similar their angles must be equal  and their sides must be proportional 



A pie chart representing the population of four cities is shown here. Read the pie chart and find the population of the city $$D$$.
1083780_944d6dfcfbbe42afa5ec344db2c86b44.png

We have $$120^{\circ}$$ as A city of 100 lakh.
means
$$\dfrac{120}{360}$$ part of full population = $$100$$ lakh.
$$\dfrac{120}{360}$$ (full pop.) = $$100$$ lakh
full population = $$300$$ lakh
angle covered by D city in plechart 
$$= 360 - 120 - 96 - 60$$
$$=84$$
i.e. $$\dfrac{84}{360} $$ of full population is in city D
population of city $$D = \dfrac{84}{360} \times 300$$ lakh
Population = $$69.99$$ lakh

1329126_1083780_ans_57c3efc555654c8db42b747285f4b18c.png


What is a regular polygon?

A polygon which has all its sides
 of equal length is a Regular polygon 
eg. Square is a Regular polygon 
Also , All the angles are also equal 
in a regular polygon








In given figure, the AB and AC of$$\Delta \,ABC$$ are produced to point E and D respectively. If bisectors BO and Co of $$\angle CBE$$ and $$\angle BCD$$ recspectively  meet at point O.
1258302_45e0ae98a19c477f8a2391aae9af8ae6.png


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In the given figures, lines $$AB$$ and $$CD$$ intersect at point $$O$$ such that $$\angle A O D+\angle B O D+\angle B O C = 300 ^ { \circ }$$. Find $$\angle A O C$$
1257600_f120cb6d28e84d7398fce6a49822090d.png

we  know that $$\underbrace{\angle AOD+\angle BOD+\angle BOC}+\angle AOC=360^o$$
                               Given $$=300^o$$

$$\Rightarrow 300^o+\angle AOC=360^o$$

$$\Rightarrow \angle AOC=60^o$$


Find the measure of each exterior angle of a regular pentagon, hexagon, heptagon, polygon of 15 sides.

The measure of each exterior angle of a regular polygon $$=\dfrac{360^o}{n}$$
$$\Rightarrow$$  The measure of each exterior angle of a regular pentagon $$=\dfrac{360^o}{5}=72^o$$
$$\Rightarrow$$  The measure of each exterior angle of a regular hexagon $$=\dfrac{360^o}{6}=60^o$$
$$\Rightarrow$$  The measure of each exterior angle of a regular heptagon $$=\dfrac{360^o}{7}=51.43^o$$
$$\Rightarrow$$  The measure of each exterior angle of a regular polygon of $$15$$ sides $$=\dfrac{360^o}{15}=24^o$$






If any straight line be drawn from the vertex of a triangle to its opposite side. prove that it will be intersected by the straight line which joints the mid-points of the opposite sides.

Given that, ABC is a triangle

Let D be the mid point of AB  and E be the midpoint of AC.

F be the mid point of BC then AF is the straight line that bisects DE at point O.

TO PROVE:- DE bisects AF

PROOF:- since D and E are the mid points of the side AB and AC respectively of $$\Delta ABC$$, then

$$DE\parallel \,BC$$ ……….   (1)

since $$DE\parallel \,BC$$, then $$DO\parallel \,BF$$  [as BF be the part of BC and DO be the

part of DE]

In $$\Delta ABF$$,  then we know that D is the mid point of AC and O be the mid point of AF.

Then, $$DO\parallel \,BF$$     (by Converse of mid point theorem)

$$\Rightarrow $$ $$AO=OF$$

Hence, $$DE$$ bisects $$AF$$.

Hence proved.


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$$AD$$ bisects $$\angle {CAD} = {(8x + 6)^ \circ}$$ and $$\angle {DAB} = {(x + 20)^ \circ}$$, what is the value of x?
1142249_e9a2e525dfca4cea81e7bbfa7527d95d.jpg

Given:$$AD$$ bisects $$\angle{CAB}$$
$$\Rightarrow\,\angle{CAD}=\angle{DAB}$$
$$\Rightarrow\,8x+6=x+20$$
$$\Rightarrow\,8x-x=20-6$$
$$\Rightarrow\,7x=14$$
$$\Rightarrow\,x=\dfrac{14}{7}=2$$
$$\therefore\,x={2}^{\circ}$$


Four angles of a polygon are $$120^o$$ each and the remaining angles are all equal to $$160^o$$ each. Find the number of sides

Let the number of sides are $$n$$
Then number of interior angles $$=n$$
Sum of the angles in a polygon  $$=(n-2)180^{\circ}$$

Four angles are $$120^{\circ}$$ and the remaining angles are all equal to $$160^{\circ}$$ each.

Then sum of the angles $$=4\times120+(n-4)160$$
$$=480+160n-640$$
$$=-160+160n$$   degree

Now,
$$-160+160n=(n-2)180$$
$$-160+160n=180n-360$$
$$-160+360=180n-160n$$
$$20n=200$$
$$n=10$$



Determine the number of sides of a polygon whose exterior and interior angles are in the ratio $$1 : 5$$.

$$\textbf{Step 1: Find interior and exterior angle using sum of interior and exterior angle.}$$

               $$\text{We know that the sum of one exterior angle and one interior angle is }180°$$
               $$\text{as they form a linear pair. Now lets say that the exterior angle is }x.$$
               $$\text{And it is given that the interior angle are in the ratio }1:5$$
               $$\therefore$$ $$\text{the interior angle measure will be }5x$$
               $$\text{Now,}$$ 
               $$\Rightarrow x+5x=180°$$
               $$6x=180°$$
               $$x=30°$$

$$\textbf{Step 2: Find the number of sides using the sum of exterior angle}$$

                $$\text{Now, the sum of exterior angle in polygon }=360°$$
                $$\text{Let the number of sides be }'n'$$
                $$\therefore n\times x=360°$$
                $$n\times 30°=360°$$
                $$n=12$$
                $$\therefore$$ $$\text{The number of sides are }12$$

$$\textbf{Hence, the answer is }12.$$



Name the interesting pairs of lines and points of intersection in the given  
1297631_2f4f9a69049b49db81630667d2fbf0e6.png

$$Pairs\,\,of\,\,lines\,\,are$$
$$ \Rightarrow \overleftrightarrow l,\overleftrightarrow p,\overleftrightarrow q\,\,and\,\,\,{\text{point}}\,\,of\,\,{\text{inter}}\,\,{\text{section}}\,\,are$$
$$A\,\,and\,\,B\,\,Ans.$$

1204684_1297631_ans_6d37de5bfb0a4eb094c829640ddc77cb.PNG


Name the intersecting pairs of lines and points of intersection in the given figure 
1297627_f2a0253dc76e481b8a5ad228fb70b0ae.png

The pair of lines are $$l$$ and $$r$$ and the point at which they intersect is $$O$$.


Fill in the blank.
The diagonals of the quadrilateral DEFG are ____ and ____.

A diagonal is a straight line connecting the opposite corners of a polygon through its vertex.
1869858_1294608_ans_f0d6cc480a624b22925d46264526fbba.jpg


How many vertices and sides do these polygons have?
Octagon.

Number of sides/vertices of octagon = 8


How many vertices and sides do these polygons have?
Hexagon.

Number of sides/vertices of hexagon = 6


In fig., name:
Point of intersection of the lines $$q$$ and $$n$$.
1319741_7ddb03699cda4ee5bf587e2cbda2ddf0.jpg

The point of intersection of the lines $$q$$ and $$n$$ is $$C$$.


In fig., name:
Point of intersection of the lines $$p$$ and $$q$$.
1319746_c7d3de653a0440a7abfe5da0faf6b818.jpg

Point of intersection of lines $$p$$ and $$q$$ is $$A$$.


Name any four polygons along with their number of sides. 

Triangle - 3 sides
Rectangle - 4 sides
Pentagon - 5 sides
Hexagon - 6 sides


In fig., name:
Point of intersection of the lines $$r$$ and $$n$$.
1319744_a93b50d113864142bb7847986d60b886.jpg

Point of intersection of lines $$r$$ and $$n$$ is $$D$$.


How many vertices and sides do these polygons have?
Pentagon.

Number of sides/vertices of pentagon = 5


The perimeter of a regular pentagon is $$100$$ cm. How long is its each side?

Regular Pentagon has 5 side and all are equal.

Therefore, 
 perimeter = sum of all the sides
perimeter = 5 × side

100 = 5 × side

side = 100 ÷ 5

side = 20cm



Draw a rough figure and label suitably in the following case.
$$\overleftrightarrow{XY}$$ and $$\overleftrightarrow{PQ}$$ intersect at M.

$$\overleftrightarrow{XY}$$ and $$\overleftrightarrow{PQ}$$ intersect at $$M$$ shown in figure.
1337644_1644295_ans_eec635dc81414a68880f25c1211627c8.png


Explain why a circle is not a polygon.

A circle is not a polygon. A polygon is a closed figure on a plane formed from a finite number of lines segments connected end-to-end. As a circle is curved, it cannot be formed from line segments, as thus does not fit the conditions needed to be a polygon.


Define a polygon.

 A polygon is a closed figure on a plane bounded by (straight) line segments as its sides


Convert $$x=\dfrac{7y+10}{4}$$ in the form of $$y=mx+c$$

$$x=\dfrac{7y+10}{4}$$

$$\Rightarrow 4x=7y+10$$

$$\Rightarrow 4x-10=7y+10-10$$

$$\Rightarrow 7y=4x-10$$

$$\Rightarrow y=\dfrac{4}{7}x-\dfrac{10}{7}$$ is of the form $$y=mx+c$$ where $$m=\dfrac{4}{7},\,\,c=\dfrac{-10}{7}$$



Examine whether the following is a polygon or not. If not, say why?
1644534_9dcb079e58694bf78af4146b3052c37b.png

No, it is not made of line segments.


Draw rough diagram to illustrate the following.
Open curve.

Step 1: To draw
(1) A rough sketch of an open curve

Step 2: Rough sketch



Here, the sketch represents an open having two endpoints.


Give reasons for the following.
Squares, rectangles, parallelograms are all quadrilaterals.

Squares, rectangles and parallelograms are closed figures with $$4$$ sides. A quadrilateral is a 4-sided closed figure. Therefore, squares, rectangles and parallelograms are all quadrilaterals.


Count the shapes in the above pictures.
1645096_585f13cd521146fd955271f8df057f64.png

-
1328373_1645096_ans_c793cf649f2b4292910314f3fb848398.png



Classify the given curves as (i) Open or (ii) Closed.
1644310_433e8426968d47829c7498274a2c4c8f.png

Step 1: To classify
(1) Whether the given curve is open or closed

Step 2: Definition
An open curve is a curve where the beginning and end points are different.
A closed curve  is a curve that joins up so that there are no end points.

Step 3: Identification based on definition
(a) Open
(b) Closed
(c) Open
(d) Closed
(e) Closed.


Name the polygon shown in the figure. Make two more examples of same type.
1644537_3dd13846d733415c9619d99ca33bfd1f.png

Step 1: To write and make
(1) Name of polygon shown in the figure
(2) Two more examples of the same type of polygon

Step 2: Name and examples
The given figure is a triangle as this is a closed figure that is made of 33 line segments.


 Two more examples of such types of the polygon are shown above.



Name the polygon. Make two more examples of same type.
1644536_fe5e58c9a561496fb4ad40b5552b8719.png

The given figure is a quadrilateral as this closed figure is made of $$4$$ line segments. Two more examples are as above.

1337400_1644536_ans_582375c03a6d48948470b45d78b051dc.png


Consider the given figure and answer the question is it a curve?
1644314_0207b15211514b929406a58bf396a379.png

Step 1: No, it is not a curve as it has no curve line in it. All the parts of figure is a straight line. So, it is a polygon not a curve.


Consider the given figure and answer whether is it closed or not?
1644315_9954d395accf4fb29ba5af705dbb1f2d.png

Step 1: Yes, it is a closed figure as it has no opening in it and It has no end points .



Examine whether the following is a polygon or not. If not, say why?
1644533_bdfbd9a0e0e44c57904c95faaaef029d.png

Yes, it is a polygon made of $$6$$ sides.


Draw a line l and a point X on it. Through X, draw a line segment $$\overline{XY}$$ perpendicular to l.
Now draw a perpendicular to $$\overline{XY}$$ at Y.(use ruler and compasses).

$$(1)$$ Draw a line l and mark a point X on it.
$$(2)$$ Taking x as centre and with a convenient radius, draw an arc intersecting the l at two points A and B.
$$(3)$$ With A and B as centre and a radius more than AX, construct two arcs intersecting each other at Y.
$$(4)$$. Join XY, $$\overline{XY}$$ is perpendicular to l.
1336414_1645104_ans_da2477efb06c4061b4fe56359a6c2fbd.png


Classify the following curve as open or closed:
1672704_3d1f4cae6ed7493da0037e22952e3d2e.png

curve is opened.as the initial and endpoints did not meet at the same end point.


Classify the following curve as open or closed:
1672701_2b1fe562fd2f4719b09c8ec1f9ef985b.png

curve is opened.as the initial and endpoints did not meet at the same end point.


Draw rough diagrams to illustrate the following:
(a) Open curve
(b) Closed curve

(a) Open Curves: Curves whose end points do not meet are called open curves.

(b) Closed curves: Curves whose end points join to enclose an area are called closed curves.

1508572_1672700_ans_32a83a6d4aeb4465bd4f96bf417b21b8.png


Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.


In order to prove that $$AB$$ and $$CD$$ bisect each other at $$O,$$ it is sufficient to prove that $$ \Delta AOD \cong \Delta BOC.$$
In these two triangles, 

we have

$$AD = BC$$

$$ \angle OBC = \angle OAD $$        [$$ \because AD \parallel BC$$ and $$AB$$ is transversal]

and, $$ \angle OCB = \angle ODA $$   [$$ \because AD \parallel BC $$ and $$CD$$ is transversal]

So, by $$ASA$$ congruence criterion, we obtain

$$ \Delta AOD \cong \Delta BOC $$

$$ \Rightarrow OA = OB $$ and $$ OD = OC $$

$$ \Rightarrow $$ $$AB$$ and $$CD$$ bisect each other at $$O$$.

1396911_1669819_ans_6b020dac08a94726b5f462be9ce9d3ce.jpg


Classify the following curve as open or closed:
1672705_4a4c2336b6e94584b5983cdf2d7c1b0c.png

curve is opened.as the initial and endpoints did not meet at the same end point.


Classify the following curve as open or closed:
1672706_3ce0600147e540da9759c66e52f78d64.png

closed curve. as the initial point and final point is same,


A polygon has $$44$$ diagonals. Find the number of its sides.

Let there be n sides of the polygon. We know that number of diagonals of n sided polygon is $$\dfrac{n(n-3)}{2}$$.

Therefore,
$$\dfrac{n(n-3)}{2}=44$$

$$n^2-3n-88=0$$

$$(n-11)(n+8)=0$$

$$n=11$$  as $$n$$ is positive

Hence, there are 11 sides of the polygon.


Given two points $$P $$ and $$Q,$$ find how many line segments do they determine?

In this problems we have given given two points $$P$$ and $$Q.$$
Now, if we try to join these two points through a line segment, we can see that there can be only one such line segment $$PQ.$$
$$\therefore$$  Given two points, only $$one$$ line segment can be determined by them.

1522616_1669220_ans_3c666f55fba841b394b5cb6c4569ff36.png


Classify the following curve as open or closed:
1672702_f9e092dd6a7c45c6b30a7c04279b2de8.png

curve is closed.as the initial and endpoints met at the same end point.


There are a number of ways by which we can visualize a portion of a line. 
Type 1 if the following statement represents a portion of a line otherwise 0.

Wire between two electric poles.


1498262_1674529_ans_d54fde0c3a094054abcd0877d80c44a3.jpg


From figure, write lines whose point of intersection is D.
1674551_d707d68e15644c5f8e87603c1b3d2e0a.png

Point of intersection of line $$r$$ and line $$l$$ is $$D.$$


From figure, write lines whose point of intersection is E.
1674552_31e4deaa6bb547ce9b8e8c10086466db.png

Point of intersection of line $$r$$ and $$m$$ is $$E.$$


Complete the following, so as to make a true statement:
A diagonal of a quadrilateral is a line segment that joins two ____ vertices of the quadrilateral.

The line segment joining the opposite vertices of quadrilateral is the diagonal of the quadrilateral.

In the given figure, the two diagonals are AC and BD.

1508670_1672761_ans_feeb4754326a49139789dfcb7a9081f2.png


Illustrate, if possible the following with a rough diagram:
A polygon with two sides.

Polygon with 2 sides
Not possible
A polygon, which is a geometric figure, has three or more sides.
1508598_1672710_ans_d5a57f2408b04ad5a259ed29dfdd47f3.png


Illustrate, if possible the following with a rough diagram:
An open curve made up entirely of line segments

An open curve made up entirely of line segments is shown in figure.

1508597_1672709_ans_e524343c2af04af1b3ef769e87173c03.png


Given are some figures. Classify each of these figures on the basis of the following:
(i) simple curve  (ii) simple closed curve  (iii) polygon  (iv) Convex polygon  (iv) convex polygon  (v) concave polygon  (vi) Not a curve
1672711_5edba92366b04fff92ae70e0f17990cb.png

$$(i)$$ Simple closed curve and a concave polygon

$$(ii)$$ Simple closed curve and a convex polygon

$$(iii)$$ Not a curve and hence it is not a polygon

$$(iv)$$ Not a curve and hence it is not a polygon

$$(v)$$ Simple closed curve

$$(vi)$$ Simple closed curve 

$$(vii)$$ Simple closed curve 

$$(viii)$$ Simple closed curve 


From figure, write all pairs of intersecting lines.
1674549_3a984aae169b4f0a9ed8ed08ae837640.png

The pairs of intersecting lines are : $$l,p\ ;\ l,q\ ;\ l,r\ ;\ m,p\ ;\ m,q\ ;\ m,r\ ;\ n,p\ ;\ n,q\ ;\ n,r\ ;p,q\ ; \ p,r$$
In total there are 11 pairs of intersecting lines.


What is a regular polygon? State the name of a regular polygon of 
(i) 3 sides  (ii) 4 sides  (iii) 6 sides


$$\textbf{Step-1: Write the definition of regular polygon & identify the polygon using given sides}$$
                 $$\text{A polygon is said to be a regular polygon, if all its interior angles are equal,}$$
                 $$\text{sides are equal & exterior angles are equal.}$$ 
                 $$\text{The name of a regular polygon of :}$$
                 $$(i)$$ $$\text{3 sides}$$ $$=\text{equilaterial triangle.}$$
                 $$(ii)$$ $$\text{4 sides}$$ $$=\text{square.}$$
                 $$(iii)$$ $$\text{6 sides}$$ $$=\text{ regular hexagon.}$$
$$\textbf{Hence, the names of a regular polygon with given sides are stated.}$$

2164087_1672715_ans_42b2d69fe19a4868ba3fe6bc226c0e7b.png


From figure, write lines whose point of intersection is $$I$$.
1674550_8ed87f63fd8d45588a97df0d8a6402a9.png

Point of intersection of line $$p$$ and $$m$$ is $$I.$$


Write the names of these lines.

$$Ref$$ Image
The names of these lines by joining A, B, C, D in pairs are $$AB,\ BC,\ CD,\ DA,\ AC$$, and $$BD$$.
1671915_1674558_ans_20c56290036e4d2f88ff2e9dc3d2a073.png


Name the lines which are concurrent at A.

$$AB,\ AC $$  and $$AD$$  are the lines concurrent at $$A$$.


From figure, write lines whose point of intersection is A.
1674553_303081d459e941689ef1de74dd4d367a.png

Point of intersection of line $$l$$ and $$q$$ is $$A.$$


Classify the following curve as open or closed.

1675154_68b67515e134477e8f73738f8e5df8d6.png

Curve is open as the initial and final points didn't meet.


Count the number of line segments in figure.

1675129_8b534e6c51954838b04ef4966d06ba6f.png

Following are the line segments given in figure:
$$AB , AC, AD, AE,$$ $$BC, BD, BE,$$ $$CD, CE,$$ $$DE$$
Therefore, the total number of line segments $$= 4 + 3 + 2 + 1 = 10$$


Draw a polygon and shade its interior. Also draw its diagonals, if any.

ABCD is a polygon and AC and BD are their diagonals
1714785_1675164_ans_c1bad6a59c17417aa337a662883f00a8.png


Write the arms and the vertex of $$\angle LMP$$ given in figure.
1675435_ca1c0acd66074d51a62935c065d3f77e.png

Arms : MP, ML
vertex : M


Is it a closed curve?
1675183_e999ac650f23481b88dce2e59d980716.png

It is a closed curve as the initial and final points met.


Is it a closed curve?
1675186_8588161f3bd84215a72f5857432101e0.png

It is a closed curve as the initial and final points met.


Is it a closed curve? 
1675180_33d250d9979848b586b41aaf655a61f6.png

It is a open curve as initial and finial points did not meet.


Is it a closed curve?


1675181_cd6b6ae347fe49069d685615b68c5686.png

It is a closed curve as initial and finial points  met.


Is it a closed curve?
1675182_41baa3cf803c4a17b5940cb811adbf75.png

It is a closed curve as initial and finial points met.


How many angles are formed in the figure. Name them.
1675438_c2a2e8f547c7438d9ec7da43a09279b2.png

3 angles are formed.
Angles are $$\angle ABC, \angle BAC,\angle  BCA..$$


Is it a closed curve?
1675184_e4893c6249374d80a5786550a35a4ef2.png

It is a open curve as the initial and final points didn't meet.


In the adjoining figure, name:
Three lines, whose point of intersecting is $$P$$
1777929_c045943b038e4757a1ed651ca4bfc65f.png

Three lines whose point of intersecting is PP


In the adjoining figure, name :
(i) Two pairs of intersecting lines and their corresponding points of intersection.
(ii) Three concurrent lines and their points of intersection 
(iii) three rays
(iv) Two line segments 
1715406_73f176ff450c4e75a6bcea0f82916415.PNG

(i)Two pairs of intersecting lines and their corresponding points of intersection :  $$ \left\{ \overleftrightarrow {EF} ,\overleftrightarrow {GH},R \right\}, \left\{ \overleftrightarrow {AB},\overleftrightarrow {CD} ,P \right\} $$

(ii)Three concurrent lines are $$\overleftrightarrow {AB}, \overleftrightarrow {EF}, \overleftrightarrow {GH}  $$ and their points of intersection is point $$R$$

(iii) Three rays : $$ \overrightarrow {RB} , \overrightarrow {RH} , \overrightarrow {RG} $$

(iv) Two line segments : $$ \overline { RQ} , \overline {RP} $$


Define the following term :
 Half-line

A straight line which extends from a point indefinitely in one direction only is as a half-line


Fill in the blank:
The curves which have different beginning and end points are called ____ curves.

The curves which have different beginning and end points are called $$\textbf{open}$$ curves.


From the given figure, write all pairs of intersection lines.
1787296_5dc957348eec47d99c8618d0ddb449a2.png

Pairs of intersecting lines are :  $$l, n\ ;\  l, p\ ;\  m, n\ ;\ m, p\ ; \ n, p$$


Draw rough diagram to illustrate the following:
open simple curve

A rough diagram of an open simple curve is shown in the image.
1720745_1787492_ans_4a79b795ab4440848b9c6fc0bf459957.png


In the given figure. Name all pairs of intersecting lines.
1787319_f5b0322960694b8bb7bf2131f29911c7.png

The pairs of intersecting lines are : $$AC, ED\ ;\ AC,AD\ ; \ AC, DC\ ;\ AD, ED\ ;\ AD, DC\ ;\ ED,DC$$ 


In the given figure, write all pairs of intersecting lines.
1787771_b4424885866a4d4486893b8900410722.png

The pairs of intersecting lines are : $$l, p\ ;\ m, p\ ;\ n,p\ ;\ l,q\ ;\ m,q\ ;\ n,q\ ;\ l,r\ ;\ m,r\ ;\ n,r\ ;\ p,q\ ;\ p,r\ ;\ q,r$$


In the given figure, write the name of concurrent lines.
1787772_f47765a49fd34d4da426423bc10584d3.png

The lines $$n,q,r$$ are concurrent at $$A$$


Examine whether the following figures are polygons. Give reasons.

1788398_00f58f2822a04c10a746ec512731093a.PNG

Polygon, because it is a simple closed curve made up entirely of line segments.


Examine whether the following figures are polygons. Give reasons.
1788397_794b6c4e666f4133b44d6595c787f144.PNG

Not a polygon, because it is not closed curve.


Name each of the following polygons:
1788403_33db7bb140bb4d6c9747da45622275c1.PNG

Pentagon


Name each of the following polygons:
1788405_4e11560aaa23406c85cf7bb764ca16f5.PNG

Hexagon


Name each of the following polygons:
1788404_5558628618774fb89de784a1ba703980.PNG

Quadrilateral


Examine whether the following figures are polygons. Give reasons.
1788399_011cd73729f64ec99b2e702f3a87c558.PNG

Not a polygon, because it is not a simple curve. i.e. the figure is intersected.


Name each of the following polygons:
1788406_7fe350658f594428be120cbb0c2bd5c2.PNG

Octagon


Examine whether the following figures are polygons. Give reasons.
1788400_5e5d0bb3d4294d17b60a390d32d0d2f1.PNG

Not a polygon, because it is not up of entirely line segment.


The common part between the two angles $$BAC$$ and $$DAB$$ in figure is ______
1788723_a14ad387e50548249bbb41078a60af43.png

$$\textbf{Finding the common part in the two angles.}$$

Q31-sol.png

From the above images of each angle separately, we can observe that in BAC and ∠DAB We have ray AB as a common part between them.




The number of common points in the two angles marked in figure is _____.
1788721_79b40f4d55564be8806ed33fa471c0dc.png

$$\textbf{Finding the common points in the two angles marked.}$$

Different ways to write names of the marked angle are,

  1. FQD And DAG so, point D is common.

  2. FQG And DAG so, point G is common.

  3. BQD And BAG so, point B is common.

  4. FQG And BAF so, point F is common.

$$\textbf{Hence, there are 4 common points in the angles marked in the given figure.}$$



The number of diagonals in a hexagon is ______

$$\textbf{Finding the number of diagonals.}$$
$$\textbf{A line segment which joins two non-adjacent vertices of a polygon is known as a diagonal.}$$
     On the basis of this definition, we will draw all the possible diagonals of a hexagon. 
             
     Now from the above figure,
$$\textbf {We can conclude that the number of diagonals in a hexagon is 9.}$$




In figure, points lying in the interior of the triangle $$PQR$$ are ____, that in the exterior are ______ and that on the triangle itself are _______
1788690_5d37afc8f359487e8eb7a00d39529ebb.png

If we have a closed figure then the points which lie within that closed figure are known as interior points; those lying outside the closed figure are known as exterior points and those lying on the closed figure are known as boundary points.
So, 
• O, S are interior points.
• T, N are exterior points.
• M is a point on the ∆PQR itself.



The number of common points in the two angles marked in figure _____
1788718_f1deab8045044a1e88bf07bce2468e70.png

$$\textbf{Finding the common points in the two angles marked.}$$

                The names of the marked angles can be written as PEQ and PAR or PEQ and   QAR  or  PER and  QAR.

                So, while writing the same angles using different points, we observe that there are 3 points P, Q and R that are common in the angles marked in the given figure.


$$\textbf{Hence, there are 3 points in common.}$$



The number of common points in the two angles marked in figure is _____
1788716_0fa225da2a33498e92a7cb46abec87f4.png

$$\textbf{Finding the common points in the two angles marked.}$$

            The name of the marked angles is BAC and DAE.

            So, while writing the names of the angles point A is used in both of them.

$$\textbf{Hence, there is only 1 common point in the marked angles in the given figure.}$$



The number of common points in the two angles marked in figure is ____
1788714_6c6fd20c8c9b41a69ab9a9926ba16450.png

$$\textbf{Finding the common point in the two angles marked.}$$

            The name of the marked angles is PAQ and PDQ. 

            So, while writing the names of the angles point P and Q is used in both of them.


$$\textbf{Hence, there are 2 common points in the marked angles in the given figure.}$$



Name the following angles of figure, using three letters:
(a) $$\angle 1$$  (b) $$\angle 2$$  (c) $$\angle 3$$  (d) $$\angle 1+\angle 1=2$$  (e) $$\angle 2+\angle 3$$  (f) $$\angle 1+\angle 2+\angle 3$$  (g) $$\angle CBA-\angle 1$$
1788750_bc316aacc74946f0bbdf01ef4af6e484.png

                 If we want to name, the angle at B it is done by writing it in the middle of  
                 points on the two rays/arms from which it has been formed.
       
i.e. ∠ABC or ∠CBA both are same.
             (a) ∠1 = ∠DBC
             (b) ∠2 = ∠EBD
             (c) ∠3 = ∠ABE
             (d) ∠1+∠2 = ∠EBC
             (e) ∠2 +∠3 = ∠ABD
             (f) ∠1+∠2+∠3 = ∠ABC
             (g) ∠CBA - ∠1 = ∠1+∠2+∠3 - ∠1
                                        = ∠2+∠3
                                        = ∠ABD




Name the line segments shown in figure.
1788744_f1872e2a1e4f4eecb1f0a40bf97f2734.png

$$ \textbf{Line segment refers to the shortest distance between two points.}$$ 
             By using the above given definition,
             we can say that in the given figure AB, BC, CD, DE, and AE are the only distinct line segments.



Name the points and then the line segments in each of the given figures.
1788751_8fd1cdbaccef4364a9ce98b0fdaef4a0.png

(i) Number of vertex are $$A,B$$ and $$C$$. Number of segment are $$AB,BC$$ and $$CA$$
(ii) Number of vertex are $$A,B,C$$ and $$D$$. Number of segment are $$AB,BC,CD$$ and $$DA$$
(iii) Number of vertex are $$A,B,C,D$$ and $$E$$. Number of segment are $$AB,BC,CD,DE$$ and $$EA$$
(iv) Number of vertex are $$A,B,C,D,E$$ and $$F$$. Number of segment are $$AB,CD$$ and $$EF$$


Name the vertices and the line segments in figure.
1788747_26f4fec78387496495eb685516dbee35.png

$$\textbf{Points at which edges of a polygon intersect or meet are known as vertices of the polygon.}$$ 
                   By using the above definition, vertices in the given figure are A, B, C, D, and E.

$$ \textbf{Line segment refers to the shortest distance between two points.}$$ 
                  Using the above definition,
                  AB, BC, CD, DE, AE, AC, AC and AD DE are the line segments 
                  present in the given figure.



Write down fifteen angles (less than $${180}^{o}$$) involved in figure.
1788748_335a70c436354bdf9bfc0411bbee7858.png

Fifteen angles which are less than 180° are,
             ∠FBC, ∠FCB, ∠BFC, ∠FEB, ∠FBE, ∠EFB, ∠FDC, ∠FCD, ∠CFD, ∠BAC,  
             ∠ABC, ∠ACB, ∠ABF, ∠ADF and ∠EFD.



Name all the line segments in figure.
1788743_b14fce7523a14db794a69aeaaaa789e2.png

$$ \textbf {Line segment refers to the distance (shortest) between two points.}$$
             Using the above definition,
             AB, AC, AD, AE, BC, BD, BE, CD, CE and DE are the line segments 
             present in the given figure.



Will the measure of $$\angle ABC$$ and of $$\angle CBD$$ make measure of $$\angle ABD$$ in figure?
1788760_99c52a0ce4da4994ac35da1360ab0996.png

In the given figure, line segment BC divides ∠ABD in two parts which are ∠ABC and ∠CBD respectively.
$$\textbf{Hence, the measure of ∠ABC and ∠CBD will definitely make measure of ∠ABD.}$$ 



If two rays intersect, will their point of intersection be the vertex of an angle of which the rays are the two sides?

               Let us consider the two rays be L and M,
     
              Q58.png
As the rays L and M intersect at O and the formation of angles takes place at O as well.
$$\textbf{Hence, O is the vertex of angle of which the rays are the two sides.}$$




In figure, 
(a) What is $$AE+EC$$?
(b) What is $$AC-EC$$?
(c) What is $$BD-BE$$?
(d) what is $$BD-DE$$?
1789081_352b530faab244268ead4f2e04f888bd.png

As points A, E and C are collinear and point E divides AC in two parts such that AE and EC are two parts.
Similarly, as points B, E and D are also collinear and point E divides BD in two parts such that BE and ED are two parts

(a). AE+EC=AC
(b). AC-EC=AE
(c). BD-BE=ED
(d). BD-DE=BE



How many points are marked in figure?
1789145_42c8abac3e114e9982add2cda675c695.png

In the given figure, A and B are the points marked.
$$\textbf{Hence, two points are marked in the given figure.}$$



Will the lengths of line segment $$AB$$ and line segment $$BC$$ make the length of line segment $$AC$$ in figure?
1788763_294f83ac81e449dc8058248323e9f073.png

In the given figure, point A, B and C are three collinear points such that point B divides line segment AC in two parts which are AB and BC respectively.

$$\textbf{Hence, the lengths of line segment AB and the line segment BC make the length of line segment AC.}$$ 



Which points in figure appear to be mid-points of the line segments? When you locate a mid-point, name te two equal line segments formed by it
1788753_fe0f244e29934b88a646f48a11bdd18d.png

(i) The given figure show there is no midpoint
(ii) the given figure show there is midpoint, $$O$$ midpoint $$AB$$ and line segment are $$AO$$ and $$OB$$
(iii) the given figure show there is a mid-point. $$D$$, midpoint $$BC$$ and line segment are $$BD$$ and $$DC$$


Is it possible for the same
(a) Line segment to have two different lengths?
(b) Angle to have two different measures?

(a) No, it is not possible for same line segment to have two different lengths
(b) No, it is not possible for same angle to have two different measures.


Find out the incorrect statement, if any, in the following:
An angle is formed when we have
(a) Two rays with a common end-point
(b) Two line segments with a common end-point
(c) A ray and a line segment with a common end-point

$$\textbf{An angle is formed when we have two rays meeting at a common point.}$$ 

In the light of the above definition, we can conclude that statements (b) and (c) are incorrect.



Look at figure and mark a point
(a) $$A$$ which is in the interior of both $$\angle 1$$ and $$\angle 2$$
(b) $$B$$ which is in the interior of only $$\angle 1$$
(c) Point $$C$$ in the interior of $$\angle 1$$
Now, state whether points $$B$$ and $$C$$ lie in the interior of $$\angle 2$$ also
1788769_ab15d102e7fb4e949cbbcc28ef5651be.png

$$\textbf{Identifying the position of points B and C.}$$ 
            In the given figure, we can observe that 2 is greater than 1 .
            Also, 2 completely overlaps 1.
            Point B and C are in the interior of 1 and 1 is completely overlapped by 2 so, automatically point B and C become interior points of 2 as well.
$$\textbf{Hence, points B and C lie in the interior of ∠2 also.}$$ 



In figure
(a) Is $$AC+CB=AB$$?
(b) Is $$AB+AC=CB$$?
(c) Is $$AB+BC=CA$$?
1788815_df9739b1cac44a2db2bd8431fc2ab1db.png

(a). As point A, C and B are collinear and point C divides line segment AB such that AC and CB are the two parts.
$$\textbf{Hence, AC+CB=AB is true.}$$

(b). From the given figure we can clearly say that AB+AC is not equal to CB.
$$\textbf{Hence, AB+AC=CB is false.}$$

(c). From the given figure we can clearly say that AB+BC is not equal to CA.
$$\textbf{Hence, AB+BC=CA is false.}$$


Draw all the diagonals of a pentagon $$ABCDE$$ and name them.

The diagonals of pentagon $$ABCDE$$ are drawn as shown in figure.

The name of the diagonals are : $$AC,AD,BE,BD$$ and $$EC$$

1637172_1789256_ans_610428d733ac4ad8ad214890f2d178b6.png


How many line segments are there in figure? Name them
1789154_993e310cd71f49a6b9900435b1d9af64.png

$$\textbf{Finding the number of line segments.}$$ 
            Shortest distance between two given points is known as the line segment.
            As there are only three points in the given figure namely A, B and C.
            Hence, AB, BC, AC are the three-line segments possible in the given figure.
$$\textbf{Hence, the total number of line segments is 3.}$$ 



In figure, how many line segments are there? Name them
1789160_d98aec562eb74fbd9060a236b0126179.png

$$\textbf{Finding the number of line segments.}$$ 
            Shortest distance between two given points is known as the line segment.
            As there are only four points in the given figure namely A, B, C and D.

           Hence, all the possible line segments in the given figure are,
           AB, AC, AD, BC, BD and CD.
$$\textbf{Hence, the total number of line segments are 6.}$$ 



In figure how many points are marked? Name them.
1789157_e381fc943a284d52a04ff82f5a31272d.png

In the figure,
Four points  $$A,B,C,D$$  are marked.


In figure, how many line segments are there? Name them.
1789164_23417e2d7c6342e79457b91a43fb7733.png

$$\textbf{Finding the number of line segments.}$$ 
            Shortest distance between two given points is known as line segment.
            As there are only five points in the given figure namely A, B, D, E and C.
            All the possible line segments in the given figure are,
            AB, AD, AE, AC, BD, BE, BC, DE, DC and EC.
$$\textbf{Hence, total number of line segments are 10.}$$ 



In figure, how many points are marked? Name them
1789153_7b2bb00fa78f43cdab49c16411194fb5.png

In the given figure, A, B and C are the only points which are marked.
$$\textbf{Hence, three points are marked in the given figure.}$$ 



The drawing below Fig., show angles formed by the goalposts at different positions of a football player.The greater the angle, the better chance the player has of scoring a goal. For example, the player has a better chance of scoring a goal from Position $$A$$ than from Position $$B$$.
In Parts (a) and (b) given below it may help to trace the diagram and draw and measure angles.
Seven football players are practicing their kicks. They are lined up in a straight line in front of the goalpost Fig. Which player has the best (the greatest) kicking angle?

1791042_b21a1aecb1cf4ad2a6cb8a64458eb999.png

$$4^{th}$$ player has the highest degree of angle to kick. Hence, this player has best kicking angle.


For each angle given below, write the name of the vertex, the names of the arms and the name of the angle. 
1821740_8c9d1173afbd4d92a26b7a9eb51a993c.png

In the given figure,
Vertex $$=O$$
Arms $$=OA$$ and $$OB$$
Angle $$=\angle AOB$$ or $$\angle BOA$$ or $$\angle O$$


Solve the following :
The ratio between exterior angle and interior angle of a regular polygon is $$ 1 : 5 $$. Find the number of sides of the polygon. 

From the question it is given that, The ratio between an exterior angle and the interior angle of a regular polygon is 1: 5
Let us assume exterior angle be $$ y $$ And interior angle be $$ 5 y $$
We know that, sum of interior and exterior angle is equal to $$ 180^{\circ} $$,
$$\begin{array}{l}y+5 y=180^{\circ} \\6 y=180^{\circ} \\y=180^{\circ} / 6 \\y=30^{\circ}\end{array}$$
the number of sides in the polygon The number of sides of a regular polygon whose each interior angles has a measure of
$$150^{\circ}$$
Let us assume the number of sides of the regular polygon be $$ n $$,
Then, we know that $$ 150^{\circ}=((2 n-4) / n) \times 90^{\circ} $$
$$\begin{array}{l}150^{\circ} / 90^{\circ}=(2 n-4) / n \\5 / 3=(2 n-4) / n\end{array}$$
By cross multiplication,$$3(2 n-4)=5 n$$
$$6 n-12=5 n$$
By transposing we get,
$$\begin{array}{l}6 n-5 n=12 \\n=12\end{array}$$
Therefore, the number of sides of a regular polygon is 12 .


Fill in the blanks to make the statements true. 
A polygon is a simple closed curve made up of only ________ . 

Line segment 


Fill in the blanks to make the statement true. 
A nonagon has _______ sides .

A nonagon has nine sides .


Fill in the blank to make the statement true. 
The name of three-sided regular polygon is _____ . 

A three sided polygon is called a triangle. Further, a triangle having all its sides equal is known as an equilateral triangle.

Hence, The name of three-sided regular polygon is $$\text{equilateral triangle.}$$


Some of the figures are given:
Classify each of them on the basis of the following:
(a) Simple curve
(b) simple closed curve
(c) Polygon
(d) Convex polygon
(e) Concave polygon
1810090_7f6dd7d0ba6c4134b0fe37b8b132cd7d.png

The given figure are classified as 
(a) Figure (i), Figure (ii), Figure (iii), Figure (v), and Figure (vi) are simple curves. A simple curve is a curve that does not cross itself. 
(b) Figure (iii), Figure (v), and Figure (vi) are simple closed curves. In simple closed curves the shapes are closed by line-segments or by a curved line. 
(c) Figure (iii) and Figure (vi) are Polygons. A Polygon is any 2-dimensional shape formed with straight lines. 
(d) Figure (iii) is a Convex polygon. In a convex polygon, every diagonal of the figure passes only through the interior points of the polygon. 
(e) Figure (vi) is a Concave polygon. In a concave polygon, at least one diagonal of the figure contains points that are exterior to the polygon.


Fill in the blanks to make the statements true. 
Is a closed curve made up of line segments  ? The name for this shape is ______ . 
1792191_06a7ff62dfe94114b30d198a1965d2eb.png

Concave Polygon 


From the  adjoining figure, write all the lines which intersect $$EF$$.
1821964_f958262f1ca042b0890a689deb26dce4.png

$$AB$$ and $$CD$$ are the lines which intersect $$EF$$


Give two examples from your surroundings for the following:
intersecting lines

The edges of textbook and notebook through any corner are intersecting lines.


State, if the following is polygon:
1823254_933b3624a1774e479ecc3b6bfc58c488.png

The given figure is closed.
Hence, the figure is a polygon.


For the angle given below, write the name of the vertex, the names of the arms and the name of the angle. 
1821741_7fe36b3d7aab4d2cb6cbc01b8e6ba0d6.png

For the angle given,
Vertex is point $$Q$$ .
Arms are $$QP$$ and $$QR$$ .
Name of the angle is $$\angle PQR$$ or $$\angle RQP$$ .
1755137_1821741_ans_9280d303a59b411fbfe5ba48c4db231e.png


State, if the following is polygon:
1823255_798582b225f64ff9958a3b17f5a52255.png

The given figure is closed.
Hence, the figure is a polygon.


For each angle given below, write the name of the vertex, the names of the arms and the name of the angle. 
Name the angles marked by letters $$a, b, c, x$$ and $$y$$
1821746_547d6a249496407c89194eea3fed45f8.png

Vertix $$=O$$
Arms $$=OA,OB,OC,OD,$$ and $$OE$$
Angles,
$$a=\angle AOE$$
$$b=\angle AOB$$
$$c=\angle BOC$$
$$d=\angle COD$$
$$e=\angle DOE$$


From the adjoining figure, write lines whose point of intersection is $$G$$.
1821970_e664b24da43e48fabd9195d092f177eb.png

The lines whose point of intersection is $$G$$ are $$AB$$ and $$GH$$.


For each angle given below, write the name of the vertex, the names of the arms and the name of the angle. 
1821742_40c99aae5b4e45e8aebed57469764a5f.png

In the given figure,
Vertex $$=M$$
Arms $$=MN$$ and $$ML$$
Angle $$=\angle LMN$$ or $$\angle NML$$ or $$\angle M$$


State, whether the above pairs of lines or rays appear to be parallel or intersecting.
1821863_28cc319895964666ab3398d4d3830cb8.png

The given line in the figure are intersecting lines.


State, if the following is polygon:
1823250_700cce0f1c3f4fd7958cda7c9a88c7bb.png

The given figure is not closed.
Hence, the figure is not a polygon.


State, if the following is polygon:
1823261_e78ff4ada0e7495fa6e51e93f564ff3e.png

The side intersects each other in the given figure.
Hence, the figure is not polygon.


The ratio between the interior angle and the exterior angle of a regular polygon is $$2:1$$. Find:
number of sides in the polygon.

Interior angle : exterior angle = 2 : 1
Let interior angle = 2x° & exterior angle = x°

∴ 2x° + x° = 180°

3x = 180°

(i)  x = 60°

∴ Each exterior angle = 60°

Let no. of sides = n

∴ 360° /n = 60°

n = 360° /60°

n = 6

(ii)

∴ (i) 60°    (ii) 6


2004068_1823353_ans_6d4e5885dd954478b11656c806f53e35.PNG


State which of the following are polygons:

1839042_64c8e9c6c43e42b1bc51a602f3712020.png


1974163_1839042_ans_cef50c49a4554eda82e5e4683b0f22f5.jpeg


State which of the following are polygons:

1839044_fafb92743fda4b8db3b517cc8abde81f.png


1974164_1839044_ans_4f6f783f35b24eb1bb9d456a199f0a60.jpeg


State which of the following are polygons:

1839037_4be880d4163740f9b8333dc09776dc89.png


1974161_1839037_ans_b11b259c0f864d69b1c0faa4eba9ccd2.jpeg


State, if the following is polygon:
1823258_e43c0aaace71443ba2ad2e90408b2eaf.png

In the given figure, one of the sides is an arc.
Hence, the figure is not polygon.


Fill in the blanks:
In case of regular polygon, with 
Number of sidesEach exterior angleEach interior angle
$$6$$.....................

Each exterior angle $$=360^o /6$$
$$=60^o$$
Each interior angle $$=180^o -60^o$$
$$=120^o$$


State which of the following are polygons:

1839036_adddac99caf24edba0a2a42631d1160f.png


1974160_1839036_ans_e3ec1f6f79714049ae6371d919877027.jpeg


State whether the given figure is polygon or not.
1839031_b3b0d7dd53c84c94aad1b479013abb88.png


1974159_1839031_ans_4fb1f38d676b44f8a89cbbfc8b377a29.jpeg


Consider the given figure and answer the questions :
a) Is it a curve ?
b) Is it closed ?
1866637_beed3e4a04154be793df35390142924c.png


2007834_1866637_ans_7b02add382084e2b99ad45f26db54672.jpg


In fif 3.42 , if lines PQ and RS intersect at point T such , that $$\angle PRT = 40 , \angle RPT = 95^{\circ} $$ and $$\angle TSQ = 75^{\circ} $$ , find $$\angle SQT $$
1869771_b83ccefc1aa24d0cb079830c9d9d742a.PNG

PQ and RS stright  lines intersect at T . If$$\angle PRT = 40^{\circ} , \angle RPT = 95 ^{\circ}$$ and $$\angle TSQ = 75 ^{\circ} $$ , then $$\angle SQT = ? $$
In$$\bigtriangleup  PRT $$
$$\angle  RPT + PRT + PTR = 180^{\circ}$$
$$95 + 40 + \angle  PTR = 180^{\circ}$$
$$135 + \angle  PTR = 180$$
$$\angle PTR = 180 - 135 $$
$$\therefore \angle PTR = 45^{\circ}$$
$$\angle PTR = \angle STQ = 45^{\circ}(\because$$ Verically opposite angles) 
In $$\bigtriangleup  TSQ $$
$$\angle STQ + \angle  TSQ + \angle SQT = 180$$
$$45 + 75 + \angle SQT = 180 $$
$$\therefore \angle SQT = 180 - 120 $$
$$\therefore \angle SQT = 60^{\circ}$$
1830430_1869771_ans_042a1babf2454f8dbb0a7ad3dcdce605.PNG


Use the figure to name:
a) Line containing point E
b) Line passing through A.
c) Line on which O lies.
d) Two pairs of intersecting lines
1866597_fa3e1e615d9a4b8eab70f1f479027286.png

a) $$\overleftrightarrow{AE}$$
b) $$\overleftrightarrow{AE}$$
c) $$\overleftrightarrow{OC}$$
d) $$\overleftrightarrow{OC}$$ and $$\overleftrightarrow{AE}, \overleftrightarrow{AE}$$ and $$\overleftrightarrow{EF}$$


Illustrate, if possible, each one of the following with a rough diagram :
a) A closed curve that is not a polygon
b) An open curve made up entirely of line segments
c) A polygon with two sides.

c) This is not possible as the polygon having the least number of sides is a triangle which has three side in it.
1956229_1866639_ans_94b85161999448c287e237b23440fa33.png


Name each polygon.
Make two more examples of each of these.
1867680_1f55f91f043f405b8fa80334f6e1e63e.png

The given figure is a Quadrilateral as this closed figure is made of $$4$$ Line segments two more example are.
1823338_1867680_ans_b78931df0122427daa2117fdede96038.png


Examine whether the following are polygons. If any one among them is no, say why?
1867669_a5ab2d874c794e21853c0179b6c0d72a.png

a) It is not a polygon as it is not a closed figure
b) Yes, it is a polygon made of $$6$$ sides.
c) No, it is not made of line segments.
d) No, it is not made of only line segments.


Classify the following curves as
(i) open or
(ii) closed
1866631_d13b21b95dd74fdf86d51a56615ced53.png

a)open curve
b)closed curve
c)open curve
d)closed curve
e)clossed curve


In fig 3.14 , lines XY and MN intersects at O . If $$\angle POY = 90^{\circ} $$ and $$a : b = 2:3 , $$ find c 

$$\angle XOP + \angle POY = 180^{\circ}$$
$$\because$$ Straight supplementary 
$$\angle XOP +90^{\circ} = 180^{\circ} $$
$$\therefore \angle XOP = 180^{\circ} - 90^{\circ} = 90^{\circ}$$
But, $$\angle XOP = a+ b = 90^{\circ}$$
$$a:b = 2:3$$
$$2+3 = 5$$ Ratio 
Ratio 5 means $$90^{\circ}$$
$$\dfrac{2\times 90}{5}=2\times 18=36^{\circ}$$
$$\therefore $$ If $$ a =36^{\circ} $$ then , $$\angle b = 54^{\circ} $$
$$\angle XOM = \angle b = 54^{\circ}=\angle YON$$$$\because $$ Vertically opposite angles ) 
$$ \angle XON + \angle YON = 180^{\circ} $$ ($$\because $$ v Straight angle ) 
$$\therefore  c + 54^{\circ} = 180^{\circ}$$
$$c = 180 - 54 $$
$$ \therefore  c = 126^{\circ}$$
1829883_1869624_ans_17cfbe5b568342449c7fca2ccaed7378.PNG


Write the answer of each of the following questions:
(iii) Write the name of the point where these two lines intrsect.

The name of the point where these two lines intersect is called origin.
Coordinates of origin are $$(0, 0)$$.


Name each polygon.
Make two more examples of each of these.
1867685_79d81b0d1e894986b66a7992a7c3c5e8.png

The given figure is an octagon as this closed figure is made of $$8$$ find segments. Two more examples are.
1823373_1867685_ans_685ce3ced5b14d5eaea1aa719964459e.png


Write the name of the point where these two lines intersect.

 Origin


The straight roads intersects each other at O. A testing center have to formed such that its distance from O be 1 km and equidistant from two roads, Show the possible cases of testing center by figure.
1876792_53a7ea8c03284b6b910f36b9ac0eb8a0.png

Possible cases are four points P, Q, R and S. In which two points P and Q lie on bisector of $$ \angle AOC $$ and two points R and S on bisector of $$ \angle BOC $$.
Thus intersecting point O lie at equidistant from the roads and on the bisector of $$ \angle AOC $$ or $$ \angle BOC. $$
1861883_1876792_ans_db6b88e980f14ee480e09485e8ca0fac.png


Find the number of sides of a regular polygon if each exterior angle is equal to its adjacent interior angle

Let $$e$$ be the exterior angle and $$i$$ be the interior adjacent angle of a regular polygon.

The sum of exterior angle and its adjacent interior is $$180^0$$, that is,
$$e + i = 180^0$$

Since each exterior angle is equal to its adjacent interior angle, therefore, substitute $$i = e$$

$$e+e=180^{ 0 }$$

$$ \Rightarrow 2e=180^{ 0 }$$ 

$$\Rightarrow e=\dfrac { 180 }{ 2 } =90^{ 0 }$$

We know that the measure of exterior angle is $$e=\left( \dfrac { 360 }{ n }  \right) ^{ 0 }$$ where $$n$$ is the number of sides of a polygon.

Here, it is given that the exterior angle is $$e=90^0$$, therefore, 

$$n=\dfrac { 360 }{ e }$$

    $$=\dfrac { 360 }{ 90 } $$
    $$=4$$

Hence, the number of sides is $$4$$.


Find the number of sides of a regular polygon if each exterior angle is equal to half its adjacent interior angle

The sum of exterior angle and its adjacent interior is $$180^0$$, that is,

$$e + i = 180^0$$
 
Since each exterior angle is equal to half its adjacent interior angle, therefore, substitute $$\dfrac { i }{ 2 }=e$$ or $$i=2e$$.

$$e+2e=180^{ 0 }\\ \Rightarrow 3e=180^{ 0 }\\ \Rightarrow e=\dfrac { 180^{ 0 } }{ 3 } =60^{ 0 }$$

We know that the measure of exterior angle is $$e=\left( \dfrac { 360 }{ n }  \right) ^{ 0 }$$ where $$n$$ is the number of sides.

Here, it is given that the exterior angle is $$e=60^0$$, therefore, 

$$n=\dfrac { 360 }{ e } =\dfrac { 360 }{ 60 } =6$$

Hence, the number of sides is $$6$$.


Find the number of sides of a regular polygon if each exterior angle is equal to one third of its adjacent interior angle.

The sum of exterior angle, $$e$$ and its adjacent interior angle, $$i$$ is $$180^\circ$$:

$$e + i = 180^\circ$$
 
Since, each exterior angle is equal to one third its adjacent interior angle, therefore, substitute $$\dfrac { i }{ 3 }=e$$ or $$i=3e$$ in the above equation.

$$\implies e+3e=180^{ \circ }\\ \\ 4e=180^{ \circ }\\ \\ e=\dfrac { 180^{ \circ } }{ 4 } =45^{ \circ }$$

We know that the measure of exterior angle is $$e=\left( \dfrac { 360 }{ n }  \right) ^{ \circ }$$ where $$n$$ is the number of sides.

Here, it is given that the exterior angle is $$e=45^\circ$$, therefore, 

$$n=\dfrac { 360^\circ }{ e } =\dfrac { 360^\circ }{ 45^\circ } =8$$

Hence, the number of sides is $$8$$.


A triangle is a closed planar shape with




2058074_1489479_ans_a255090280f8484aa6634dc96b6ec57d.jpg


A variable straight line of slope $$4$$ intersects the hyperbola $$xy = 1$$ at two points. Find the locus of the point which divides the line segment between these two points in the ratio $$1:9$$.


2041535_1051528_ans_b9754cde3ad44181b40f4f9fa6f5ada1.jpg


In fig., name:
Point of intersection of the lines $$l$$ and $$m$$.
1319739_d78624e7467c49778b37798624681e32.jpg

The point of intersection of the lines $$l$$ and $$m$$ is $$B$$.


Is it possible to have a regular polygon whose each interior angle is :
$$138^{o}$$

Suppose polygon has $$n$$ side 
The sum of interior angles 
$$=(n-2)\times 180^o$$
each exterior angles is $$=\dfrac {(n-2)\times 180}{n}$$
So $$n\times 45^o$$
$$\therefore 180\,\,\, n-360=45n$$
$$135\ n=360\,\,\,\,or \,\,\,n=\dfrac {360^o}{135}=\dfrac {8}{3}$$
As $$'n'$$ should be $$+ve$$ integrate, so it is not possible to have such a polygon.


Show that the lines $$\vec{r} = (3\hat{i} + 2\hat{j} - 4\hat{k})$$ + $$\lambda(\hat{i} + 2\hat{j} + 2\hat{k})$$ and $$\vec{r} = (5\hat{i} - 2\hat{j})$$ + $$\mu(3\hat{i} + 2\hat{j} + 6\hat{k})$$ are intersecting and find their point of intersection. 


2042815_1534916_ans_9338eca346814f87a0920853b5440794.png


Classify the following curve as open or closed.
1675158_eeb9d731cc81405b80ccb4249942b3bd.png

Open


Which of the following are closed curves? Which of them are simple?
1675185_d796d83f7601465c9672c5817b904dc3.png

It is a closed curve as the initial and final points met.


Class 6 Maths Extra Questions