Question 1: In the given figure, A, B and C are three points on a circle with centre O such that ∠BOC = 30° and ∠AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.

**Answer:** ∠AOC = 60° + 30° = 90°

Since angle made by an arc on the centre is double the angle made anywhere else on the circle

Hence, ∠ADC = 90°/2 = 45°

Question 2: A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

**Answer:** Given a circle with centre O in which chord AB = radius OB

As OA = OB = AB

So, ΔOAB is equilateral triangle

So, ∠AOB = 60°

Angle made on centre by an arc is double the angle made anywhere else on the circle

Hence, ∠ADB = 30°

As ACBD is a cyclic quadrilateral

Hence, ∠ACB = 180°- ∠ADB

Or, ∠ACB = 180°- 30° = 150°

Question 3: In the given figure, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.

**Answer:** Here, PQRS is a cyclic quadrilateral

Hence, ∠PSR = 180° - ∠PQR

Or, ∠PSR = 180° - 100° = 80°

As angle made by an arc on centre is double than angle made anywhere else

So, ∠POR = 2 x 80° = 160°

In ΔPOR we have PO = RO

So, ∠OPR = ∠ORP

Or, 160° + 2∠OPR = 180°

Or, 2∠OPR = 180° - 160°= 20°

Or, ∠OPR = 10°

4. In the given figure, ∠ABC = 69°, ∠ACB = 31°, find ∠BDC.

**Answer:**In ΔABC;

∠BAC = 180° - (69° + 31°)

Or, ∠BAC = 180° - 100° = 80°

Angles made on one side of a chord are equal

Hence, ∠BDC = ∠BAC = 80°

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