Question

prove that in a parallelogram the opposite sides are equal

Answer 1

Let ABCD be a parallelogram, withA = α an B = β. Prove thatC = α andD = β.α + β= 180°(co-interior angles, AD || BC),so C= α (co-interior angles, AB || DC)and D= β (co-interior angles, AB || DC).

Second property of a parallelogram − The opposite sides are equal

As an example, this proof has been set out in full, with the congruence test fully developed. Most of the remaining proofs however, are presented as exercises, with an abbreviated version given as an answer

The opposite sides of a parallelogram are equal.

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.