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In a Cyclic-trapezium, the Non-parallel Sides Are Equal and the Diagonals Are Also Equal. Prove It. - Mathematics

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Question

In a cyclic-trapezium, the non-parallel sides are equal and the diagonals are also equal. Prove it.

Solution


A cyclic trapezium ABCD in which AB ∥ DC and AC and BD are joined.
To prove-
(i) AD = BC
(ii) AC = BD
Proof :
∵ chord AD subtends ∠ABD and chord BC subtends BDC
At the circumference of the circle.

But ∠ABD = ∠BDC   [proved]
Chord AD = Chord BC
⇒ AD = BC    

Now in ∆ADC and ∆ BCD

DC = DC      [Common]

∠CAD = ∠CBD           [angles in the same segment]

And  AD = BC        [proved]

By Side – Angle – Side criterion of congruence, we have

∴ ∆ADC ≅ ∆BCD       [ SAS axion]

The corresponding parts of the congruent triangle are congruent

 ∴ AC = BD       [c.p.c.t]

  Is there an error in this question or solution?
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APPEARS IN

 Selina Solution for Concise Mathematics for Class 10 ICSE (2020 (Latest))
Chapter 17: Circles
Exercise 17(B) | Q: 1 | Page no. 265
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In a Cyclic-trapezium, the Non-parallel Sides Are Equal and the Diagonals Are Also Equal. Prove It. Concept: Arc and Chord Properties - Angles in the Same Segment of a Circle Are Equal (Without Proof).
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