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If Two Sides of a Cyclic Quadrilateral Are Parallel; Prove That: (I) Its Other Two Sides Are Equal. (Ii) Its Diagonals Are Equal. - ICSE Class 10 - Mathematics

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Question

If two sides of a cyclic quadrilateral are parallel; prove that:
(i) its other two sides are equal.
(ii) its diagonals are equal.

Solution

Given –
ABCD is a cyclic quadrilateral in which AB ∥ DC. AC and BD are its diagonals.
To prove –
(i) AD = BC
(ii) AC = BD
Proof –

(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]

  Is there an error in this question or solution?
Solution If Two Sides of a Cyclic Quadrilateral Are Parallel; Prove That: (I) Its Other Two Sides Are Equal. (Ii) Its Diagonals Are Equal. Concept: Arc and Chord Properties - Angles in the Same Segment of a Circle Are Equal (Without Proof).
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