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 DC and AD BC Now, AB DC and transversal AD intersects them at A and D respectively. <A + <D = 180 ....(i) [ Sum of consecutive interior angles is 180] Again, AD BC and DC intersects them at D and C respectively. <C + <D = 180 ...(ii) [ Sum of consecutive interior angles is 180] From (i) and (ii), we get <A + <D = <D + <C <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 and To Prove: ABCD is a Parallelogram. Proof: In a quadrilateral ABCD .......(1) [Given] .......(2) [Given] Adding (1) and (2) .............(3) Now [Angle Sum Property] Using equation (3) we get But and are cointerior angles when AD and BC are straight lines and AB is the transversal cutting them As their sum is Therefore AD || BC .......(4) Again and is given But and are cointerior angles when AB and CD are straight lines and BC is the transversal cutting them As their sum is 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 sum property of triangle.] 5x+ 20+ 2x+ 10 +3x= 180 10x+ 30 = 180 10x = 180-30=150 x= 15 In a parallelogram sum of adjacent angles= 180 In a parallelogram opposite angles are equal and Hence |
In the following figure, ABDC, ABFE are parallelograms. Which of the following is true about the angles of this parallelogram? | |||
Right Option : D | |||
View Explanation |
In the following figure, If ABCD is a parallelogram, then which angles are equal? | |||
Right Option : D | |||
View Explanation |
In the following figure, which of the angles will be equal if PQRS is a parallelogram? | |||
Right Option : B | |||
View Explanation |
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