PR and QS are diameters of circle T. What is the measure of SR?
50°, 80°, 100°, 120°

Solution:

It is given that

∠PQT = 40°

The radii of the circle are QT and PT

QT = PT

Triangle PQT is an isosceles triangle

∠PQT =∠QPT = 40°

Triangle RST is an isosceles triangle

RST = ∠SRT

In triangle PQT and SRT

ST ≅ TQ (radii of the circle)

PT ≅ TR (radii of the circle)

Using the vertical theorem

∠PQT ≅ ∠STR

Triangle PQT ≅ triangle RST (SAS congruence theorem)

By cpct

∠PQT = ∠RST = 40° (from (1))

∠QPT = ∠SRT = 40° (from (2))

In ΔSTR,

Sum of all the angles = 180°

40° + 40° + m∠PTQ = 180°

m∠STR = 180° - 80°

m∠STR = 100°

We know that

The measure of a central angle and the arc it intercepts are equal in measure.

m∠STR = m(arc SR) = 100°

Therefore, the measure of SR is 100°.

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PR and QS are diameters of circle T. What is the measure of SR?

Summary:

PR and QS are diameters of circle T. The measure of SR is 100°.