Question

pr and qs are diameters of circle t. what is the measure of sr? 80° 100° 120° 50°

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. In \(\triangle PQT\), \(\angle PQT = 40^{\circ}\), and \(\angle PQT\) is an inscribed angle that intercepts arc \(PT\). So the measure of arc \(PT\) is \(2\times\angle PQT=80^{\circ}\).

Step2: Use properties of a circle

Since \(PR\) is a diameter of circle \(T\), the measure of arc \(PQR\) is \(180^{\circ}\). Also, since \(QS\) is a diameter, the circle is symmetric about \(QS\).
The measure of arc \(PR = 180^{\circ}\). We know that arc \(PR=\text{arc }PT+\text{arc }TR\), and \(\text{arc }PT = 80^{\circ}\), so \(\text{arc }TR=180 - 80=100^{\circ}\).

Step3: Find the measure of arc \(SR\)

The measure of arc \(SR\) is \(100^{\circ}\) because vertical - angles formed by the intersection of two diameters in a circle result in equal - measure arcs.

Answer:

\(100^{\circ}\)