pq is the diameter.if angle pqr =65,angle rps=25 and angle qpt=60 then find angle rpq,prs,psr and pqt

The given information can be represented as,

Given: ∠pqr = 65°, ∠rps = 25° and ∠qpt = 60°

∠prq is an angle in a semicircle.

∴ ∠prq = 90°     [Angle in a semicircle is 90º]

In Δpqr,

∠pqr + ∠prq + ∠rpq = 180°      [Angle sum property]

⇒ ∠rpq = 180° – (90° + 65°) = 25°

In cyclic quadrilateral pqrs,

∠pqr + ∠psr = 180°         [ Sum of opposite angles in cyclic quadrilateral is 180°]

⇒ ∠psr = 180° – 65° = 115°

In Δprs,

∠prs + ∠psr + ∠rps = 180°         [Angle sum property]

⇒ ∠prs = 180° – (115° + 25°) = 40°

∠ptq = 90°      [Angle in a semicircle is 90º]

In Δptq,

∠ptq + ∠pqt + ∠qpt = 180°          [Angle sum property]

⇒ ∠pqt = 180° – (90° + 60°)

⇒ ∠pqt = 180° – 150°

⇒ ∠pqt = 30°

∴ ∠rpq = 25°, ∠prs = 40°, ∠psr= 115° and ∠pqt = 30°