Use circle theorems relating tangents and chords (alternate segment theorem).\nPQ is a tangent to a circle at point T. Points R and S lie on the circle such that TR = TS, and ∠RST = 65 degrees. What is the value of angle ∠PTS?

Introduction / Context:
This problem combines two classic geometry ideas: (1) isosceles triangles formed by equal chords or equal segments in a circle, and (2) the tangent-chord theorem (also called the alternate segment theorem). The goal is to find the angle between a tangent at T and the chord TS, which equals the angle in the opposite segment subtended by chord TS.

Given Data / Assumptions:

• PQ is tangent to the circle at T.
• R and S are points on the circle.
• TR = TS, so triangle RST is isosceles with equal sides meeting at T.
• Angle RST = 65 degrees.
• We need angle PTS (angle between tangent PT and chord TS).

Concept / Approach:
First use isosceles triangle properties: if TR = TS, then base angles at R and S are equal. Next apply the tangent-chord theorem: the angle between the tangent at T and chord TS equals the angle in the alternate segment, which is the angle subtended by chord TS at the opposite point on the circle (here point R). So ∠PTS equals ∠TRS.

Step-by-Step Solution:
Given TR = TS, triangle RST is isosceles with base RS So base angles are equal: ∠TRS = ∠RST Given ∠RST = 65°, therefore ∠TRS = 65° Now use tangent-chord (alternate segment) theorem: Angle between tangent at T and chord TS equals the angle in the opposite segment subtended by TS That opposite-segment angle is ∠TRS So ∠PTS = ∠TRS = 65°

Verification / Alternative check:
We can also find ∠RTS = 180° - 65° - 65° = 50°, consistent with an isosceles triangle. The tangent-chord theorem does not depend on this 50°, but it supports internal consistency of the given triangle angles.

Why Other Options Are Wrong:
130° is the exterior angle at S if incorrectly doubled, but ∠PTS is not defined that way.
115° and 55° come from mixing up which angle corresponds to the alternate segment.
45° is an unrelated guess and does not match any derived circle angle relationship here.

Common Pitfalls:
Confusing the tangent-chord theorem with the angle at the center theorem, or using the wrong point for the “alternate segment” angle. Another frequent mistake is forgetting that TR = TS implies angles at R and S are equal, not angles at T and S.

Final Answer:
∠PTS = 65°