Question

The given figure shows a circle with centre O and ∠ABP = 42°.

Calculate the measure of:

1. ∠PQB
2. ∠QPB + ∠PBQ

Answer 1

In the figure

∠ABP = 42°.

Join PO, QO

∵ Arc PA subtends ∠POA at the centre and

∵ ∠PBA at the remaining part.

∴ ∠POA = 2∠PBA = 2 × 42° = 84°

But ∠AOP + ∠BOP = 180°  ...(Linear pair)

`\implies` ∠POA + ∠POB = 180°

`\implies` 84° + ∠POB = 180°

`\implies` POB = 180° – 84° = 96°

Similarly, arc BP subtrends ∠BOP on the centre and ∠PQB at the remaining part of the circle

∴ `∠PQB = 1/2`

`∠POB = 1/2 xx 96^circ = 48^circ`

But in ΔABQ,

∠QPB + ∠PBQ + ∠PQB = 180°   ...(Angles of a triangle)

∠QPB + ∠PBQ + 48° = 180°

`\implies` ∠QPB + ∠PBQ = 180°

`\implies` ∠QPB + ∠PBQ = 180° – 48° = 132°

Hence i. PQB = 48° and ii. ∠QPB + ∠PBQ = 132°