Question

in circle p, the measure of ∠qps is 15° and rq is a diameter. find the arc measures.

Explanation:

Step1: Recall central - angle and arc - measure relationship

The measure of an arc is equal to the measure of its central angle.

Step2: Find \(m\widehat{QS}\)

Since \(\angle QPS = 15^{\circ}\), then \(m\widehat{QS}=15^{\circ}\).

Step3: Find \(m\widehat{ST}\)

We know that \(\angle SPT = 105^{\circ}\), so \(m\widehat{ST}=105^{\circ}\).

Step4: Find \(m\widehat{TR}\)

Since \(RQ\) is a diameter, the sum of the arcs of a semi - circle is \(180^{\circ}\). Let \(m\widehat{TR}=x\). We have \(m\widehat{QS}+m\widehat{ST}+m\widehat{TR}=180^{\circ}\). Substituting the known values: \(15^{\circ}+ 105^{\circ}+x = 180^{\circ}\), then \(x=180^{\circ}-(15^{\circ}+105^{\circ}) = 60^{\circ}\).

Answer:

\(m\widehat{QS}=15^{\circ}\)
\(m\widehat{ST}=105^{\circ}\)
\(m\widehat{TR}=60^{\circ}\)