Question

If two consecutive angles of cyclic quadrilateral are congruent, then prove that one pair of opposite sides is congruent and other is parallel.

Answer 1

Given: ABCD is a cyclic quadrilateral and ∠ABC ≅ ∠BCD.

To prove: Side DC ≅ Side AB, AD || BC

Construction: Draw seg AM and seg DN both perpendicular to side BC.

Proof: ∠ABC ≅ ∠BCD  ......(i) [Given]

∠ABC + ∠ADC = 180°  ......(ii) [Opposite angles of a cyclic quadrilateral are supplementary]

From equations (i) and (ii),

∠BCD + ∠ADC = 180°

∴ Side AD || Side BC  .....[Interior angles test]

In ΔDNC and ΔAMB,

seg DN ≅ seg AM  .......[Perpendicular distance between two parallel lines]

∠DNC ≅ ∠AMB   ......[Each is 90°]

∠DCN ≅ ∠ABM  ......[Given]

As a result, the SAA test of congruence

ΔDNC ≅ ΔAMB

∴ Side DC ≅ Side AB  ......[C.S.C.T.]

Hence, side AD || side BC and side DC ≅ side AB. 

Hence proved.