If two sides of a cyclic quadrilateral are parallel; prove that:
(i) its other two sides are equal.
(ii) its diagonals are equal.
ABCD is a cyclic quadrilateral in which AB ∥ DC. AC and BD are its diagonals.
To prove –
(i) AD = BC
(ii) AC = BD
(i) AB || DC ⇒ ∠DCA = ∠CAB [Alternate angles]
Now, chord AD subtends ∠DCA and chord BC subtends ∠CAB
At the circumference of the circle.
∴ ∠DCA = ∠CAB [proved]
∴ Chord AD = Chord BC or AD = BC
(ii) Now in ∠ABC and ∠ADB ,
AB = AB [Common]
∠ACB = ∠ADB [Angles in the same segment]
BC = AD [Proved]
By Side – Angle – Side criterion of congruence, we have
ΔACB ≅ ΔADB [SAS postulate]
The corresponding parts of the congruent triangles are congruent.
∴ AC = BD [c.p.c.t]
Video Tutorials For All Subjects
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