Line P R and Line Q S are diameters of circle T. What is the measure of Arc S R?

50°
80°
100°
120°

Option C:

m(arc SR) = 100°

Solution:

∠PQT = 40°

QT and PT are radii of circle.

QT = PT

ΔPQT is an isosceles triangle.

⇒ ∠PQT = ∠QPT = 40° – – – (1)

Similarly, ΔRST is an isosceles triangle.

⇒ RST = ∠SRT – – – (2)

In ΔPQT and ΔRST,

PT ≅ TR (Radii of circle)

ST ≅ TQ (Radii of circle)

∠PQT ≅ ∠STR (by vertical theorem)

Hence ΔPQT ≅ ΔRST by SAS congruence theorem.

By corresponding parts of congruence triangles are congruent.

∠PQT = ∠RST = 40° (by (1))

∠QPT = ∠SRT = 40° (by (2))

In ΔSTR, sum of all the angles = 180°

40° + 40° + m∠PTQ = 180°

m∠STR = 180° – 80°

m∠STR = 100°

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

m∠STR = m(arc SR)

m(arc SR) = 100°

Hence option C is the correct answer.