**Telangana**** SCERT Solution Class IX (9) Math Chapter 12 Circles Exercise 12.5**

**(1) (i) Solution:**

From the adjacent figure sides opposite to ∠x & ∠y are equal

∠x = ∠y [∵opposite angles of equal sides a triangle are equal]

∴ x + y + 30^{o} = 180^{o} [sum of interior angles of triangle is 180^{o}]

Or, x + x + 30^{o} = 180^{o} [∵x = y]

Or, 2x = 180^{o} – 30^{o}

Or, x = 150/2 = 75^{o}

∴ x = y = 75^{o}

∴ x = 75^{o}, y = 75^{o}

**(ii)** From the adjacent figure

y + 85^{o} = 180^{o}

y = 180^{o} – 85^{o} = 95^{o}

x + 110^{o} = 180^{o}

Or, x = 180^{o} – 110^{o}

= 70^{o}

∴ x = 70^{o}

y = 95^{o}

Since sum of opposite site angles of a cycle quadrilateral is 180^{o}

**(iii)** From the adjacent figure,

x = 90^{o} [∵the triangle is a right angles triangle at x]

y + x + 50^{o} = 180^{o} [∵sum of interior angles of a is 180^{o}]

Or, y + 90^{o} + 50^{o} = 180^{o}

Or, y = 180^{o} – 140^{o}

= 40^{o}

x = 90^{o}, y = 40^{o}

**(2) Given that the vertices A, B, C of a quadrilateral ABCD lie on a circle. Also ****∠****A + ****∠****C = 180°, then prove that the vertex D also lie on the same circle**

**Solution:**

Given,

That In quad ABCD, AB lie on an circle

∠A + ∠C = 180^{o}

∴ ∠A & ∠C are opposite angles of quad ABCD

Also, ∠A + ∠C = 180^{o} therefore they are supplementary

∴ Quad ABCD is a cyclic quadrilateral

∴ All the vertices A, B, C, D lie on the circle

**(3) If a parallelogram is cyclic, then prove that it is a rectangle**

**Solution:**

Given, ABCD is a cyclic parallelogram

∴ ∠A = ∠C [opposite angles of a ∥gm are equal]

Also,

∠A + ∠C = 180^{o} [∵ opposite angles of cyclic quadrilateral are supplementary]

Or, ∠A + ∠A = 180^{o} [<A** = **<C]

Or, ∠A = 180^{o}

Or, ∠A = 90^{o} = ___

∴ ||gm ABCD is a rectangle since opposite angles are 90^{o}

**(4) Prove that a cyclic rhombus is a square.**

**Solution:**