PQ is a diameter of a circle with centre O. RS is a chord parallel to PQ and subtends an angle of 40° at the centre O. If PR and QS are produced to meet at T, what is the measure (in degrees) of angle ∠PTQ?

Introduction / Context:
This geometry problem involves a circle, a diameter, a chord parallel to the diameter and the angle formed by intersecting secants. It tests your ability to combine properties of central angles, chords and the angle between two lines that intersect outside the circle.

Given Data / Assumptions:

• PQ is a diameter of the circle with centre O.
• RS is a chord parallel to PQ.
• Chord RS subtends an angle of 40 degrees at the centre O, so angle ROS = 40 degrees.
• Segments PR and QS are extended to meet at point T outside the circle.
• We need to find angle PTQ.

Concept / Approach:
There is a known theorem about the angle between two secants that intersect outside a circle. If two secants from an external point T cut the circle at points P, R and Q, S, then: ∠PTQ = 1 / 2 (difference of measures of intercepted arcs) In this case, arcs intercepted by the two secants are major arc PQ and minor arc RS (or vice versa). Once we know the measures of these arcs in degrees, we can apply the secant angle formula.

Step-by-Step Solution:
Step 1: Since RS subtends a central angle of 40 degrees at O, the minor arc RS also has measure 40 degrees. Step 2: PQ is a diameter, so arc PQ is a semicircle with measure 180 degrees. Step 3: Because RS is a chord parallel to diameter PQ and the geometry is symmetrical, the secants PR and QS will intercept arc PQ and arc RS in a way that allows use of the exterior secant angle formula. Step 4: For two secants TPQ and TRS intersecting the circle at P, Q and R, S respectively, the angle at the external point T is given by: ∠PTQ = 1 / 2 (measure of far arc − measure of near arc). Step 5: Here, the far arc is the semicircular arc PQ with measure 180 degrees, and the near arc is the smaller arc RS with measure 40 degrees. Step 6: So ∠PTQ = 1 / 2 (180 − 40) = 1 / 2 (140) = 70 degrees.

Verification / Alternative check:
If drawn accurately, the configuration shows that T lies on the perpendicular bisector of segment OS, and symmetry makes the intersection angle at T acute and larger than 60 degrees but less than 90 degrees. A value of 70 degrees fits this visual intuition. Also, using coordinate geometry with a unit circle and appropriate points for P, Q, R and S confirms the computed angle as 70 degrees.

Why Other Options Are Wrong:
55 and 60 degrees: These do not match the exact formula 1 / 2 (180 − 40) and would come from incorrect use of average or misinterpreting which arcs are involved.
90 degrees: This would require the difference of arc measures to be 180 degrees, which is not the case here.
50 degrees: This would correspond to using 1 / 2 (180 − 80) or another incorrect combination of arcs.

Common Pitfalls:
The main pitfall is confusing the sum and difference of arcs in the secant angle theorem, or using the wrong arcs (for example, using the major arc corresponding to RS instead of the minor arc). Another mistake is treating T as if it were on the circle rather than outside. Carefully identifying which arcs are intercepted by each secant and applying the correct formula avoids these errors.

Final Answer:
The measure of angle ∠PTQ is 70 degrees.