Question

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

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

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(arc  PQR)`   ...[Inscribed angle theorem]

∴ `110^circ = 1/2 m(arc  PQR)`

∴ 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°]

∴ ∠PQR + x + x = 180°

∴ 70° + 2x = 180°

∴ 2x = 180° – 70°

∴ 2x = 110°

∴ `x = 110^circ/2`

∴ x = 55°

∴ ∠PRQ = ∠QPR = 55°   ...(i)

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

∴ `55^circ = 1/2 m(arc  QR)`

∴ m(arc QR) = 110°