Question

In the adjoining figure, PQRS is cyclic, side PQ ≅ side RQ, ∠PSR = 110°. 

Find 

i. measure of ∠PQR 

ii. m (arc PQR) 

iii. m (arc QR)

iv. measure of ∠PRQ

Answer 1

i. PQRS is a cyclic quadrilateral. [Given]

∴ ∠PSR + ∠PQR = 180° [Opposite angles of a cyclic quadrilateral are supplementary]

∴ 110° + ∠PQR = 180° 

∴ ∠PQR = 180° – 110° 

∴ m ∠PQR = 70°

ii. ∠PSR = 1/2 m (arcPQR) [Inscribed angle theorem]

110° = 1/2 m (arcPQR)

∴ m(arc PQR) = 220°

iii. In ∆PQR, 

side PQ ≅ side RQ [Given] 

∴ ∠PRQ = ∠QPR [Isosceles triangle theorem] 

Let ∠PRQ = ∠QPR = x 

Now, ∠PQR + ∠QPR + ∠PRQ = 180° [Sum of the measures of angles of a triangle is 180°]

But, ∠QPR = 1/2 m(arc QR) [Inscribed angle theorem]

∴ 55° = 1/2 m(arc QR)

∴ m(arc QR) = 110°

iv. ∠PRQ = ∠QPR =55° [From (i)] 

∴ m ∠PRQ = 55°