Question

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

1. measure of ∠PQR
2. m(arc PQR)
3. m(arc QR)
4. measure of ∠PRQ

Answer 1

(1) ▢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°

(2) ∠PSR = `1/2` m(arc PQR)     .....[Inscribed angle theorem]

∴ 110° = `1/2` m(arc PQR)

∴ m(arc PQR) = 220°

(3) 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°]

∴ ∠PQR + x + x = 180°

∴ 70° + 2x = 180°

∴ 2x = 180° − 70°

∴ 2x = 110°

∴ x = `(110°)/2`

∴ x = 55°

∴ ∠PRQ = ∠QPR = 55°    ......(i)

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

∴ 55° = `1/2` m(arc QR)

∴ m(arc QR) = 110°

(4) In ∆PQR,

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

70° + ∠PRQ +  55° = 180°

∠PRQ = 180° - 125°

∠PRQ = 55°.