Question

Write a proof to show the opposite angles of a parallelogram are congruent. Be sure to create and name the appropriate geometric features. this figure does not need to be submitted. Give reasons

Answer 1

We have given a parallelogram ABCD

By the property of parallelogram
`AB\parallel CD`

And
`BC\parallel AD`

Prove: m∠A=m∠C

And m∠B=m∠D

If we join the diagonal AC we will get ΔABC and ΔADC

By alternative interior angles theorem ∠CAB =∠ACD

And by the same theorem ∠DAC=∠ACB

AC=AC(Reflexive)

By ASA Congruency the ΔABC≅ΔADC

By CPCTC m∠B=m∠D

Similarly, If we join diagonal BD we will get ΔABD≅ΔCBD

BY CPCTC m∠A=m∠C

Hence, Opposite angles of a parallelogram are congruent.

Answer 2

Explanation:

Let ABCD be the parallogram.

A,B,C and D are vertices in that order.

Since ABCD is a parallelogram, we get opposite sides are parallel

AB || CD

AD is a transversal making interior angles as A and D

By parallel lines properties we have A +D =180...i

Similarly since BC||AD, and AB is a transversal

Angle A+B =180...ii

We have from i and ii, Angle B = angle D

On the same grounds, as above, we can prove that

Angle A = Angle C

It follows that in a parallelogram opposite angles are equal.