Solve any three sub-questions:

In the following figure, secants containing chords RS and PQ of a circle intersects each other in point A in the exterior of a circle. If m(arc PCR) = 26°, m(arc QDS) = 48°, then find:

(i) m∠PQR

(ii) m∠SPQ

(iii) m∠RAQ

The figure is as follows:

Given:

m(arc PCR) = 26°

m(arc QDS) = 48°

(i) To find: m∠ PQR

Inscribed Angle Theorem is stated as follows:

The measure of an inscribed angle is half the measure of the intercepted arc.

Here the inscribed angle is PQR and the intercepted angle is PCR

So by inscribed angle theorem,

m∠ PQR = 1/2 m(arc PCR)

m∠ PQR = 1/2 × 26°

m∠ PQR = 13°

(ii) To find: m∠ SPQ

Here the inscribed angle is SPQ and the intercepted angle is SDQ

So by inscribed angle theorem,

∠SPQ = 1/2 m(arc SDQ)

= 1/2 × 48°

∠SPQ = 24°

(iii) To find: m∠ RAQ

∠SPQ ≅ ∠SRQ as these angles are inscribed in the same arc

∴ the ∠SRQ = 24°

Also the ∠SRQ is the exterior angle of the ∆ARQ

∠SRQ = ∠RAQ + ∠RQA ……. (Remote Interior Angles Theorem)

24° = ∠RAQ + 13°

∴ ∠RAQ = 11°