Opposite Angles of a Parallelogram are Equal

Concept Explanation

Opposite Angles of a Parallelogram are Equal

Theorem 3: The opposite angles of a parallelogram are equal. GIVEN A parallelogram ABCD To prove <A = <C and <B = <D Proof Since ABCD is a parallelogram. Therefore, AB $large parallel$ DC and AD $large parallel$ BC Now, AB $large parallel$ DC and transversal AD intersects them at A and D respectively. $large therefore$ <A + <D = 180 ....(i) [ $large because$ Sum of consecutive interior angles is 180] Again, AD $large parallel$ BC and DC intersects them at D and C respectively. $large therefore$ <C + <D = 180 ...(ii) [ $large because$ Sum of consecutive interior angles is 180] From (i) and (ii), we get <A + <D = <D + <C $large Rightarrow$ <A = <C Similarly, <B = <D. Hence, <A = <C and <B = <D
Converse Theorem: A quadrilateral is a parallelogram if its opposite angles are equal
Given : A quadrilateral ABCD in which $angle A = angle C$ and $angle B = angle D$ To Prove: ABCD is a Parallelogram. Proof: In a quadrilateral ABCD $angle A = angle C$ .......(1) [Given] $angle B = angle D$ .......(2) [Given] Adding (1) and (2) $angle A +angle B = angle C +angle D$ .............(3) Now $angle A +angle B + angle C +angle D= 360^0$ [Angle Sum Property] Using equation (3) we get $Rightarrow ;;;angle A +angle B + angle A +angle B= 360^0$ $Rightarrow ;;;2(angle A +angle B)= 360^0$ $Rightarrow ;;;angle A +angle B= 180^0$ $Rightarrow ;;;angle A +angle B=angle C +angle D= 180^0$ But $angle A$ and $angle B$ are cointerior angles when AD and BC are straight lines and AB is the transversal cutting them As their sum is $180^0$ Therefore AD \|\| BC .......(4) Again $;;;angle A +angle B= 180^0$ and $angle A = angle C$ is given $Rightarrow ;;;angle C +angle B= 180^0$ But $angle C$ and $angle B$ are cointerior angles when AB and CD are straight lines and BC is the transversal cutting them As their sum is $180^0$ Therefore AB \|\| CD .......(5) From (4) and (5) AD \|\| BC and AB \|\| CD Hence ABCD is a a parallelogram
Illustration: Find all the angles of the parallelogram ABCD the figure given. Solution: In triangle BCD $angle B +angle C+angle D= 180^0$ {angle sum property of triangle.] 5x+ 20+ 2x+ 10 +3x= 180 10x+ 30 = 180 10x = 180-30=150 x= 15 $angle B = 5x+ 20 = 5(15)+20= 75+ 20= 95^0$ $angle C = 2x+ 10 = 2(15)+10= 30+ 10= 40^0$ $angle D = 3x = 3(15)= 45^0$ In a parallelogram sum of adjacent angles= 180 $angle D + angle C= 180^0$ $angle D +40^0= 180^0Rightarrow angle D = 140^0$ In a parallelogram opposite angles are equal $angle A = angle C = 40^0$ and $angle B = angle D = 140^0$ Hence $angle A = 40^0 , angle B = 140^0, angle C = 40^0 ; and ;angle D = 140^0$

Sample Questions

(More Questions for each concept available in Login)

Question : 1

In the following figure, ABFE is a parallelogram. Which of the following is true about the angles of this parallelogram?
A. , $angle B=angle C,angle A=angle D$		B. $angle C=angle A$ , $angle B=angle F$
C. $angle B=angle E$ , $angle A=angle F$		D. none of these
Right Option : C
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Question : 2

Which of the following statement is true about the figure given below?
A. $angle A=angle D$ , $angle B=angle C$	B. $angle A=angle F$ , $angle B=angle E$	C. $angle D=angle F$ , $angle C=angle E$	D. All of these
Right Option : D
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Question : 3

Which of the following is not true about the following figure?
A. $angle F=angle D,angle E=angle H$	B. $angle E=angle A,angle G=angle D$	C. $angle F=angle C,$ $angle A=angle D$	D. $angle C=angle A,angle B=angle D$
Right Option : C
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Linear Equations in One Variable

Introduction to Trigonometry

Introduction to Euclids Geometry

Herons Formula Complete

Polynomial I

Surface Area and Volume II

Surface Area and Volume I

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Constructions

Areas Of Parallelograms And Triangles II

Areas Of Parallelograms And Triangles I

Linear Equations in Two Variables

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Opposite Angles of a Parallelogram are Equal

Opposite Angles of a Parallelogram are Equal

Students / Parents Reviews [10]

Hiya Gupta

Shivam Rana

Om Umang

Yatharthi Sharma

Shreya Shrivastava

Prisha Gupta

Ayan Ghosh

Aman Kumar Shrivastava

Barkha Arora

Cheshta