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Question
If two consecutive angles of cyclic quadrilateral are congruent, then prove that one pair of opposite sides is congruent and other is parallel.
Solution
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.
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Solution:
MRPN is cyclic
The opposite angles of a cyclic square are `square`
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