Question

circle t has diameters rp and qs. the measure of ∠rtq is 12° less than the measure of ∠rts. what is the measure of (widehat{qp})? 78° 84° 88° 96°

Explanation:

Step1: Note the relationship between angles

Since $\overline{RP}$ and $\overline{QS}$ are diameters of circle $T$, $\angle RTS$ and $\angle RTQ$ are supplementary, that is $\angle RTS+\angle RTQ = 180^{\circ}$. Let $\angle RTS=x$. Then $\angle RTQ=x - 12^{\circ}$.

Step2: Solve for $\angle RTS$

Substitute into the supplementary - angle equation: $x+(x - 12^{\circ})=180^{\circ}$. Combine like terms: $2x-12^{\circ}=180^{\circ}$. Add $12^{\circ}$ to both sides: $2x=192^{\circ}$. Divide both sides by 2: $x = 96^{\circ}$, so $\angle RTS = 96^{\circ}$.

Step3: Recall the central - angle and arc relationship

The measure of an arc is equal to the measure of its central angle. The central angle of arc $\widehat{QP}$ is $\angle RTQ$. Since $\angle RTQ=\angle RTS - 12^{\circ}$, and $\angle RTS = 96^{\circ}$, then $\angle RTQ=84^{\circ}$. So the measure of arc $\widehat{QP}$ is $84^{\circ}$.

Answer:

$84^{\circ}$