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

Explanation:- 

(1) Given that PQRS is a cyclic quadrilateral.

• From the opposite angles of a cyclic quadrilateral are supplementary property, we have angle PSR + angle PQR = 180 degrees.
• We know that angle PSR is 110 degrees, so angle PQR = 180 – 110 = 70 degrees.

(2) In the quadrilateral PQRS,

• The angle PSR is equal to half the measure of the arc PQR.
• We know that angle PSR is 110 degrees, so the measure of arc PQR is 2 * 110 = 220 degrees.

(3) In triangle PQR,

• PQ is congruent to RQ, which means that the two sides have the same length.
• From the isosceles triangle theorem, we have angle PRQ = angle QPR.
• Let the measure of angle PRQ (or QPR) be x.
• From the sum of the measures of angles of a triangle property, we have angle PQR + angle QPR + angle PRQ = 180 degrees.
• We know that angle PQR is 70 degrees, so 70 + 2x = 180.
• Solving for x, we get x = 55 degrees.
• From the inscribed angle theorem, we know that angle QPR is equal to half the measure of arc QR.
• We know that angle QPR is 55 degrees, so the measure of arc QR is 2 * 55 = 110 degrees.

(4) In triangle PQR,

• Using the sum of the measures of angles of a triangle property, we have angle PQR + angle PRQ + angle QPR = 180 degrees.
• We know that angle PQR is 70 degrees and angle QPR is 55 degrees, so 70 + angle PRQ + 55 = 180.
• Solving for angle PRQ, we get angle PRQ = 55 degrees.