Question

if ABCD is a quadrilateral in which AB|| CD and AD=BC then prove that angle A =angleB

Answer 1

Given: ABCD is a quadrilateral AB||CD and AD = BC.

To prove: ∠A = ∠B
Construction: Draw AP⊥CD and BQ⊥CD.

Proof: In ∆ADP and ∆BCQ,
AD = BC (given)
AP = BQ (distance between the parallel sides is equal)
∠APD = ∠BQC (each is 90°)
So, ∆ADP ≅ ∆BCQ (RHS congruence criterion)
⇒ ∠ADP = ∠BCQ (C.P.C.T)
∠A + ∠ADP = 180° (Sum of adjacent interior angle is supplementary)
∠B + ∠BCQ = 180° (Sum of adjacent interior angle is supplementary)
⇒∠A + ∠ADP = ∠B + ∠BCQ
⇒∠A = ∠B (Since ∠ADP = ∠BCQ)