Question

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

Answer 1

A cyclic trapezium ABCD in which AB || DC and AC and BD are joined.

To prove:

1. AD = BC
2. 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 axiom]

The corresponding parts of the congruent triangle are congruent

Therefore, AC = BD    ...[c.p.c.t]