Question

How to prove that in a parallelogram opposite angles are equal?

Answer 1

To prove: ∠A = ∠C and ∠B = ∠D

Proof:

AB||CD and AC is the transversal

Hence ∠1 = ∠4 → (1)

ΔABC ≅ ΔADC (SSS Congruence rule)

Hence ∠3 = ∠4 → (2) and ∠1 = ∠2 → (3)

Consider, ∠A = ∠3 + ∠4

= ∠4 + ∠4 [From (2)]

= 2∠4

= 2∠1

= ∠1 + ∠2

= ∠C

Therefore, ∠A = ∠C

Similarly, we can prove ∠B = ∠D

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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.