Theorem 8.2 : In a parallelogram, opposite sides are equal Class 9

Given:  A parallelogram ABCD with diagonal AC.

To Prove: ΔABC ≅ ΔADC

Proof: In ΔABC and ΔCDA, 

BC ||AD and AC is a transversal.

So, ∠BCA = ∠DAC (Pair of alternate angles)

Also, AB || DC and AC is a transversal.

So, ∠BAC = ∠DCA (Pair of alternate angles)

and AC=CA (Common)

So, ΔABC ≅ ΔADC (ASA rule)

or, diagonal AC divides parallelogram ABCD into two congruent triangles ABC and CDA.

So, AB=DC and AD=BC (CPCT)