In the given figure ‘O’ is the centre of the circle. PQ is a tangent to the circle at B and AB = AC. If ∠CBQ = 40°, find the unknown angles x, y, z and w.

Given,

∠CBQ = 40°

In a circle, the angle between a tangent and a chord through the point of contact is equal to the angle in the opposite (alternate) segment of the circle.

∠BAC = ∠CBQ = 40°

x = 40°

Since, AB = AC.

∠ABC = ∠BCA [Angles opposite to equal sides of a triangle are equal]

In triangle ABC,

∠ABC + ∠BAC + ∠BCA = 180°

2∠ABC + 40° = 180°

2∠ABC = 180° - 40°

2∠ABC = 140°

∠ABC = $\dfrac{140°}{2}$

∠ABC = 70°.

We know that,

The angle which an arc subtends at the center is double that which it subtends at any point on the remaining part of the circumference.

∠BOC = 2∠BAC

y = 2x

y = 80°.

In triangle OBC,

OB = OC (Radii of same circle)

∠OBC = ∠OCB (Angles opposite to equal sides in a triangle are equal)

By angle sum property of triangle,

∠OBC + ∠OCB + ∠BOC = 180°

2∠OBC + 80° = 180°

2∠OBC = 100°

∠OBC = 50°

From figure,

w = ∠ABC - ∠OBC = 70° - 50° = 20°.

We know that,

Sum of opposite angles of a cyclic quadrilateral is 180°.

In cyclic quadrilateral ABCD,

∠ABC + ∠ADC = 180°

70° + z = 180°

z = 180° - 70° = 110°.

Hence, x = 40°, y = 80°, z = 110°, w = 20°.