Question

If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
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Answer 1

Answer:

Given:

ABCD is a trapezium where non-parallel sides AD and BC are equal.

Construction:

DM and CN are perpendicular drawn on AB from D and C respectively.

To prove:

ABCD is cyclic trapezium.

Proof:

In △DAM and △CBN,

AD=BC ...Given

∠AMD=∠BNC ...Right angles

DM=CN ...Distance between the parallel lines

△DAM≅△CBN by RHS congruence condition.

Now,

∠A=∠B ...by CPCT

Also, ∠B+∠C=180° ....Sum of the co-interior angles

⇒∠A+∠C=180°

Thus, ABCD is a cyclic quadrilateral as sum of the pair of opposite angles is 180°.