In the adjoining figure, PS is the bisector of ∠QPR and PT ⟂ QR and ∠Q > ∠R,…

To find the measure of angle TPS in triangle PQR, we can use the geometric properties of the altitude and the angle bisector.

Step-by-Step Solution
Understand the setup: In triangle PQR, PT is the altitude to side QR, and PS is the bisector of angle QPR. We are given that angle Q > angle R. We need to find angle TPS.

Angle Sum Property: In triangle PQR, the sum of the angles is 180°. Therefore, angle QPR = 180° - (angle Q + angle R).

Angle Bisector: Since PS is the angle bisector of angle QPR, angle QPS = angle RPS = (1/2) * angle QPR = 90° - (1/2) * (angle Q + angle R).

Right-Angled Triangle: In the right-angled triangle PTQ, angle TPQ = 90° - angle Q.

Calculate angle TPS: Angle TPS is the difference between angle QPS and angle TPQ:
Angle TPS = angle QPS - angle TPQ
Angle TPS = [90° - (1/2) * (angle Q + angle R)] - [90° - angle Q]
Angle TPS = 90° - (1/2) * angle Q - (1/2) * angle R - 90° + angle Q
Angle TPS = (1/2) * angle Q - (1/2) * angle R
Angle TPS = (1/2) * (angle Q - angle R)