Question

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.

Answer 1

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 circle's opposite (alternative) segment.

∠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 = `(140°)/2`​

∠ABC = 70°

We know that,

The angle that an arc subtends at its center is double as large as the angle that it subtends at any point along its remaining 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 the angle sum property of a triangle,

∠OBC + ∠OCB + ∠BOC = 180°

2∠OBC + 80° = 180°

2∠OBC = 100°

∠OBC = `(100°)/2`

∠OBC = 50°

From the figure,

w = ∠ABC − ∠OBC

= 70° − 50°

= 20°

We know that,

The 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°.