Question

if in a cyclic quadrilateral ABCD the opposite side AB parallel to CD. prove AC parallel to BD​

Answer 1

Step-by-step explanation:

Given: ABCD is a cyclic quadrilateral and AD || BC.

To prove: AB = CD

Construction: Draw AE ⊥ BC and DF ⊥ BC.

Proof: AD || BC

∴ ∠ADC + ∠DCF = 180° (sum of adjacent interior angles is 180°)

∠ABE + ∠ADC = 180° (sum of opposite angles of cyclic quadrilateral is 180°)

⇒ ∠ADC + ∠DCF = ∠ABE + ∠ADC

⇒ ∠DCF = ∠ABE

In DABE and DDCF, we have

∠ABE = ∠DCF (proved)

∠AEB = ∠DFC (90°)

AE = DF (distance between the parallel sides is same)

∴ DABE ≅ DDCF (AAS congruence criterion)

⇒ AB = CD (C.P.C.T)