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CBSE Questions for Class 8 Maths Understanding Quadrilaterals Quiz 10 - MCQExams.com
CBSE
Class 8 Maths
Understanding Quadrilaterals
Quiz 10
A diagonal of a paralleogram divides it into two congruent triangles.
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True
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False
Explanation
Let us consider a parallelogram $$ABCD$$ with $$AC$$ as its diagonal.
To prove: $$\triangle ABC \cong \triangle ADC$$
Proof:
We know that the opposite sides of a parallelogram are parallel.
Hence $$AB \parallel DC$$ and $$AD \parallel BC$$
$$AC$$ is the transversal of the parallel lines
$$AB$$ and $$DC$$
$$\angle BAC =\angle DCA$$ (Alternate angles) $$...(1)$$
$$AC$$ is the transversal of the parallel lines
$$AD$$ and $$BC$$
$$\angle DAC =\angle BCA$$ $$...(2)$$
In $$\triangle ABC$$ and $$\triangle ADC,$$
$$\angle BAC =\angle DCA$$
$$AC$$ is the common side
$$\angle DAC =\angle BCA$$
Therefore, $$\triangle ABC \cong \triangle ADC$$
State, with reason, whether the following statement is true or false:Every parallelogram is a rhombus.
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0%
True
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False
Explanation
Every parallelogram is not a rhombus as not all the sides are equal. However, the converse is true where every rhombus is a parallelogram
If the quadrilateral $$ABCD$$ is a parallelogram. What is the value of $$x$$?
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$$45^{\circ}$$
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$$30^{\circ}$$
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$$36^{\circ}$$
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$$37^{\circ}$$
Explanation
In parallelogram
$$ABCD,$$
We know that the o
pposite angles of a parallelogram are
equal.
Hence
$$\angle A=\angle C $$
$$\Rightarrow 3{y} = 3x + {3}$$
$$...(1)$$
and
$$\angle B=\angle D$$
$$\therefore\ \angle D = 2y - {5} $$ $$...(2)$$
We know that the sum of angles of a parallelogram is $$360^\circ$$
$$\Rightarrow (3y)+(3x+3)+(2y-{5})+(2y-{5})={360}$$
$$\Rightarrow 3y+3y+4y-10={360}$$
$$\Rightarrow 10y-{10}={360}$$
$$\Rightarrow 10y={360}+{10}$$
$$\Rightarrow 10y={370}$$
$$\Rightarrow y=\dfrac{370}{10} = {37^\circ}$$
Substitute $$y=37^\circ$$ in $$(1)$$
$$\Rightarrow 3\times {(37^\circ)} = 3x + {3^\circ}$$
$$\Rightarrow 111=3x+3$$
$$\Rightarrow 3x=111-3$$
$$\Rightarrow 3x=108$$
$$\Rightarrow x = \dfrac{{108}}{3} = 36^\circ$$
In parallelogram $$ABCD$$, the bisectors of $$\angle A$$ and $$\angle B$$ intersect at $$M$$.If $$\angle A=80^{o}$$, then $$\angle AMB=$$.........
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$$40^{o}$$
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$$50^{o}$$
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$$80^{o}$$
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$$90^{o}$$
Explanation
Given in parallelogram $$ABCD, \; AM$$ and $$BM$$ are bisectors of $$\angle{A}$$ and $$\angle{B}$$ respectively.
As we know that sum of adjacent angles in a parallelogram is $$180°$$.
$$\therefore \angle{A} + \angle{B} = 180°$$
$$\Rightarrow 2 \angle{BAM} + 2 \angle{ABM} = 180°$$
$$\Rightarrow \angle{BAM} + \angle{ABM} = \cfrac{180°}{2} = 90° ..... \left( 1 \right)$$
Now, in $$\triangle{AMB}$$,
$$\angle{BAM} + \angle{AMB} + \angle{ABM} = 180°$$
$$\Rightarrow \angle{AMB} + 90° = 180°$$
$$\Rightarrow \angle{AMB} = 180° - 90° = 90°$$
Measures of opposite angles of parallelogram are $$(3x-2) ^{\circ} $$ and $$(50-x) ^{\circ} $$. Find the measure of its other angles.
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$$143$$
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$$153$$
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$$163$$
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$$173$$
Explanation
Here, $$ABCD$$ is a parallelogram
Let $$\angle A=(3x-2)^o$$ and $$C=(50-x)^o$$
We know that in parallelogram opposite angles are equal.
$$\therefore$$ $$\angle A=\angle C$$
$$\Rightarrow$$ $$3x-2=50-x$$
$$\Rightarrow$$ $$4x=52$$
$$\Rightarrow$$ $$x=13$$
$$\Rightarrow$$ $$\angle A=(3x-2)^o=(3\times 13-2)^o=37^o$$
In parallelogram sum of adjacent angles are supplementary.
$$\therefore$$ $$\angle A+\angle B=180^o$$
$$\Rightarrow$$ $$37^o+\angle B=180^o$$
$$\therefore$$ $$\angle B=143^o$$
Hence other two equal opposite angles are $$143^o$$
Diagonals of a parallelogram ABCD intersect at point O. If $$\angle BOC = 90^\circ $$ and $$\angle BDC = 50^\circ $$, then $$\angle OAB$$ is :
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$$90^\circ $$
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$$50^\circ $$
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$$40^\circ $$
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$$10^\circ $$
A quadrilateral with two pairs of opposite sides parallel, is a :
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a rhombus
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a kite
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a trapeziun
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a parallellogram
What type of angle is formed at the point of intersection of the angle bisectors of two adjacent angle of a parallelogram?
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Acute angle
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Complementary angle
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Right angle
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Obtuse angle
Say True or False.
All the sides of a rhombus are of equal length.
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True
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False
Explanation
True
All sides of rhombus are equal length.
Say True or False.
All the sides of a parallelogram are of equal length.
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0%
True
0%
False
Explanation
In parallelogram,
Opposite sides are equal is length.
Therefore in parallelogram ABCD
$$.AB=CD$$
$$.BC=AD$$
$$.But AB\neq BC\neq CD\neq AD$$
The diagonals do not necessarily intersect at right angles in a
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parallelogram
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rectangle
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rhombus
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kite
Explanation
The diagonals do not necessarily intersect at right angles in a parallelogram. But opposite sides and opposite angles should be equal.
Two adjacent angles of a parallelogram are $$\left(2x+25\right)^{o}$$ and $$\left(3x-5\right)^{o}.$$ The value of $$x$$ is
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$$28$$
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$$32$$
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$$36$$
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$$42$$
Explanation
We know that sum of adjucent angled of a parallelogram is $$180^o$$
$$(2x+25)^o+(3x-5)^o=180^o$$
$$5x+20=180^o$$
$$5x=180^o - 20^o$$
$$5x=160^o$$
$$x=160^o/5=32^o$$
Which of the following is a property of a parallelogram ?
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Opposite sides are parallel.
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The diagonals bisect each other at right angles.
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The diagonals are perpendicular to each other.
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All angles are equal.
In a parallelogram PQRS , if $$ \angle\, P = 60^{\circ} $$, then other three angles are:
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$$ 45^{\circ} , 135^{\circ} , 120^{\circ} $$
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$$ 60^{\circ} , 120^{\circ} , 120^{\circ} $$
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$$ 60^{\circ} , 135^{\circ} , 135^{\circ} $$
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$$ 45^{\circ} , 135^{\circ} , 135^{\circ} $$
Explanation
Given: $$ \angle\, P = 60^{\circ} $$
$$ \therefore \, \angle\, R = 60^{\circ} $$ $$\therefore $$ opposite angles are equals
Similarly,
$$\angle Q = \angle S$$
Now,
$$\angle P+\angle Q = 180^o$$ {
supplementary angles because these interior angles lie on the same side of the transversal}
Therefore,
$$ \angle\, Q = \angle S = 180^o - 60^{\circ} $$
$$ \angle\, Q = \angle S = 120^o $$
If two adjacent angles of a parallelogram are in the ratio $$ 2 : 3 $$ , then the measure of angles are:
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$$ 72^{\circ} , 108^{\circ} $$
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$$ 36^{\circ} , 54^{\circ} $$
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$$ 80^{\circ} , 120^{\circ} $$
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$$ 96^{\circ} , 144^{\circ} $$
Explanation
Let the angles be $$ 2x , 3x $$
$$ 2 x + 3 x = 180 $$
$$ \Rightarrow \, 5x = 180 $$
$$ \Rightarrow \, 5x = 180 $$
So , the angles are
$$ 2 \times 36 = 72 $$
$$ 3 \times 36 = 108 $$
The angle between the two altitudes of a parallelogram through the same vertex of an obtuse angle of the parallelogram is $$ 30^{\circ} $$. The measure of the obtuse angle is
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$$ 100^{\circ} $$
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$$ 150^{\circ} $$
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$$ 105^{\circ} $$
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$$ 120^{\circ} $$
Explanation
ABCD is a parallelogram.
From the question it is given that, ∠EBF =$$ 30^o$$
Sum of interior angles of a quadrilateral = $$360^o$$
Then, ∠EBF + ∠BED + ∠EDF + ∠DFB = $$360^o$$
∠EDF = $$360^o – (90^o + 90^o + 30^o) $$
∠EDF = $$150^o $$ which is an obtuse angle.
State whether the statements are true (T) or (F) false:
The sum of interior angles and the sum of exterior angles taken in an order are equal in case of quadrilateral only.
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True
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False
Explanation
Sum of Interior angles of Polygon $$ = 180(n - 2) $$
Sum of Exterior angles of Polygon $$ = 360 $$
$$ n = 4 $$
Sum of Interior angles of Polygon $$ = 180(4 - 2) = 360^{\circ} $$
Sum of Exterior angles of Polygon $$ = 360 $$
State whether the statements are true (T) or (F) false.
If diagonals of a quadrilateral bisect each other , it must be a parallelogram.
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True
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False
State whether the statements are true (T) or (F) false.
If opposite angles of a quadrilateral are equal , it must be a parallelogram.
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True
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False
Explanation
Let's assume $$ \angle $$ A = $$ \alpha $$ and $$ \angle $$ B = $$ \beta $$
We need to prove that $$ \angle $$ C = $$ \alpha $$ and $$ \angle $$ D = $$ \beta $$
$$ \alpha + \beta = 180^{\circ} $$ (co-interior angles , AD || BC
$$ \angle $$ C = $$ \alpha $$ (Co-interior angles , AB || DC )
$$ \angle $$ D = $$ \beta $$ (co-interior angles , AB || DC ).
All parallelograms having equal areas have same perimeters.
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0%
True
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False
Explanation
False.
Area of the parallelogram is equal to product of base and height. So if area of two parallelograms are equal it does not mean their perimeter is same.
Their corresponding height and sides may be different.
Boundaries of surfaces are:
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surfaces
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curves
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lines
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points
Explanation
Boundaries of surfaces are curves.
Hence, (b) is the correct answer.
The sum of interior angles of a hexagon is:
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$$ 720^{\circ} $$
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$$ 360^{\circ} $$
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$$ 540^{\circ} $$
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$$ 1080^{\circ} $$
Explanation
In $$n$$ -sided polygon,
sum of interior angles $$=(n-2)\times180^\circ$$
Hexagon has 6 side
$$n=6$$
sum of interior angles of a hexagon $$=(6-2)\times180^\circ= 720^{\circ} $$
If the two adjacent angles of a parallelogram are $$(3x-20^\circ)$$ and $$(50^\circ -x)$$, then the value of $$x$$ is:
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$$55^\circ$$
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$$75^\circ$$
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$$20^\circ$$
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$$80^\circ$$
Explanation
We know that sum of adjacent angles of a parallelogram is $$180^o$$.
Therefore,
$$(3x-20^o)+(50^o-x)=180^o$$
$$2x+30^o=180^o$$
$$2x=150^o$$
$$x=75^o$$
Option B is correct.
Select which curves are open.
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In a parallelogram $$ABCD$$, if $$AB = 2x + 5, CD = y + 1, AD = y + 5$$ and $$BC = 3x - 4$$, what is the ratio of $$AB$$ and $$BC$$?
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$$71 : 21$$
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$$12 : 11$$
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$$31 : 35$$
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$$4 : 7$$
Explanation
In parallelogram, opposite sides are parallel and equal
$$\therefore AB=CD$$ and $$AD=BC$$
$$\Rightarrow 2x+5=y+1\longrightarrow 1$$
$$\Rightarrow 3x-4=y+5\longrightarrow 2$$
Equation 2-equation 1
$$\left( 3x-2x \right) -4-5=0+4$$
$$x=13$$
Put value of x in 1
$$2(13)+5-1=y$$
$$y=30$$
$$\cfrac { AB }{ BC } =\cfrac { 2x+5 }{ 3x-4 } =\cfrac { 2(13)+5 }{ 3(13)-4 } $$
$$\cfrac { AB }{ BC } =\cfrac { 31 }{ 35 } $$
$$\therefore AB:BC=31:35$$
If an angle of a parallelogram is four-fifths of its adjacent angle, what are the angles of the parallelogram?
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$$60^o, 120^o, 60^o, 120^o$$
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$$90^o, 90^o,90^o,90^o$$
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$$80^o, 100^o,80^o,100^o$$
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$$30^o, 150^o, 30^o, 150^o$$
In the diagram, KLMN is a constructed parallelogram.
Find the value of $$\angle KLM$$
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$$15^0$$
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$$30^0$$
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$$45^0$$
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$$60^0$$
The parallelogram PQRS is formed by joining together four equilateral triangles of side 1 unit, as shown in the figure.
What is the length of the diagonal SQ?
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$$\sqrt{7}$$
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$$\sqrt{8}$$
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$$\sqrt{6}$$
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$$\sqrt{5}$$
Which of the following statement(s) is/are correct ?
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A parallelogram in which two adjacent angles are equal is a rectangle.
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A quadrilateral in which both pairs of opposite angles are equal is parallelogram
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In a parallelogram the number of acute angles is two.
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All of these
If ABCD is a parallelogram, then $$\angle A - \angle C = \underline{ }$$
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$$180^0$$
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$$0^0$$
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between $$10^0$$ & $$180^0$$
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None of these
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