Question

In the given figure, PQ is a diameter of a circle with centre O. If ∠PQR = 65°, ∠SPR = 40° and ∠PQM = 50°, find ∠QPR, ∠QPM and ∠PRS.

Answer 1

In △ PQR

We know that PQ is the diameter

So we get

∠PRQ = 90^o as the angle in a semicircle is a right angle

Using the angle sum property

∠QPR + ∠PRQ + ∠PQR = 180^o

By substituting the values

∠QPR + 90^o + 65^o = 180^o

On further calculation

∠QPR = 180^o – 90^o – 65^o

By subtraction

∠QPR = 180^o – 155^o

So we get

∠QPR = 25^o

In △ PQM

We know that PQ is the diameter

So we get

∠PMQ = 90^o as the angle in a semicircle is a right angle

Using the angle sum property

∠QPM + ∠PMQ + ∠PQM = 180^o

By substituting the values

∠QPM + 90^o + 50^o = 180^o

On further calculation

∠QPM = 180^o – 90^o – 50^o

By subtraction

∠QPM = 180^o – 140^o

So we get

∠QPM = 40^o

Consider the quadrilateral PQRS

We know that

∠QPS + ∠SRQ = 180^o

It can be written as

∠QPR + ∠RPS + ∠PRQ + ∠PRS = 180^o

By substituting the values

25^o + 40^o + 90^o + ∠PRS = 180^o

On further calculation

∠PRS = 180^o – 25^o – 40^o – 90^o

By subtraction

∠PRS = 180^o – 155^o

So we get

∠PRS = 25^o