Question

In a parallelogram opposite sides are equal.

Answer 1

Given,
A parallelogram abcd
To prove,
AB=CD,AD=BC
Construction,
Join AC
Proof,
In triangle ADC and triangle ABC
Angle ACD = Angle CAB(alternate interior angle)
angle CAD=angle ABC(-------||-----)
AC=AC (common side)
by ASA
Triangle ADC congruent triangle ABC
therefore, AB=CD(cpct)
AD=BC (cpct).

Answer 2

Statement : In a parallelogram, opposite angles are equal.

Given : Parallelogram ABCD

To prove ; ∠A = ∠C and ∠B = ∠D

Proof :

In parallelogram ABCD,
Consider,

AD || BC and AB is transversal

∠A + ∠B = 180° [Co - int. Angles]...... (i)

Now, consider AB || DC and BC transversal

∠B + ∠C = 180° [Co - int. Angles]...... (ii)

From (i) and (ii) we get ;

∠A + ∠B = ∠B + ∠C
∠A = ∠C
∠B = ∠D

Hence, it is proved.