Question

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

Answer 1

Given: ABCD is a quadrilateral such that AB || DC and AD = BC

Construction: Extend AB to E and draw a line CE parallel to AD.

Proof: Since, AD || CE and transversal AE cuts them at A and E, respectively.

∴ ∠A + ∠E = 180°  ....[Since, sum of cointerior angles is 180°]

⇒ ∠A = 180° – ∠E  ...(i)

Since, AB || CD and AD || CE

So, quadrilateral AECD is a parallelogram.

⇒ AD = CE 

⇒ BC = CE   ...[∵ AD = BC, given]

Now, in ΔBCE 

CE = BC  ...[Proved above]

⇒ ∠CBE = ∠CEB   ...[Opposite angles of equal side are equal]

⇒ 180° – ∠B = ∠E   ...[∵ ∠B + ∠CBE = 180°]

⇒ 180° – ∠E = ∠B  ...(ii)

From equations (i) and (ii),

∠A = ∠B

Hence proved.