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

**Solution:**

We know that, if the sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.

Draw a trapezium ABCD with AB || CD

AD and BC are the non-parallel sides that are equal. AD = BC. Draw AM ⊥ CD and BN ⊥ CD.

Consider ΔAMD and ΔBNC

AD = BC (Given)

∠AMD = ∠BNC (90°)

AM = BN (Perpendicular distance between two parallel lines is same)

By RHS congruence, ΔAMD ≅ ΔBNC.

Using CPCT, ∠ADC = ∠BCD.....(1)

∠BAD and ∠ADC are on the same side of transversal AD.

∠BAD + ∠ADC = 180°

∠BAD + ∠BCD = 180° [From equation(1)]

This equation proves that the sum of opposite angles is supplementary. Hence, ABCD is a cyclic quadrilateral.

**Video Solution:**

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

### Maths NCERT Solutions Class 9 - Chapter 10 Exercise 10.5 Question 8:

**Summary:**

If the non-parallel sides of a trapezium are equal, then it is a cyclic quadrilateral