Question

ABCD is a quadrilateral in which AB || DC and AD = BC. Prove that ∠A = ∠B and ∠C = ∠D.

Answer 1

According to the question,

We have,

Quadrilateral ABCD

AB||CD and AD = BC.

To prove: ∠A = ∠B and ∠C = ∠D.

Construction: Draw DP ⊥ AB and CQ ⊥ AB.

Proof: In ΔAPD and ΔBQC,

Since ∠1 and ∠2 are equal to 90^o

∠1 = ∠2

Distance between parallel line,

AB = BC [Given]

By RHS criterion of congruence,

We have

ΔAPD ≅ ΔBQC [CPCT]

∠A = ∠B

Now, DC||AB

Since, sum of consecutive interior angles is 180^o

∠A+∠3 =180 …(1)

And,

∠B +∠4 =180 …(2)

From equations (1) and (2),

We get

∠A + ∠3 = ∠B + ∠4

Since, ∠A = ∠B,

We have,

⇒ ∠3 = ∠4

⇒ ∠C = ∠D

Hence, proved.