Question

circle t has diameters $overline{rp}$ and $overline{qs}$. the measure of $angle rtq$ is $12^circ$ less than the measure of $angle rts$.
what is the measure of $overset{\frown}{qp}$?
$78^circ$
$84^circ$
$88^circ$
$96^circ$

Explanation:

Step1: Define variables for angles

Let $m\angle RTS = x$, then $m\angle RTQ = x - 12^\circ$.

Step2: Use linear pair property

$\angle RTS$ and $\angle RTQ$ are supplementary (form a straight line).
$x + (x - 12^\circ) = 180^\circ$

Step3: Solve for $x$

$2x - 12^\circ = 180^\circ$
$2x = 192^\circ$
$x = 96^\circ$

Step4: Find $m\angle RTQ$

$m\angle RTQ = 96^\circ - 12^\circ = 84^\circ$

Step5: Find vertical angle $\angle STP$

$\angle STP = \angle RTQ = 84^\circ$ (vertical angles are congruent)

Step6: Find $m\angle QTP$

$\angle STP$ and $\angle QTP$ are supplementary.
$m\angle QTP = 180^\circ - 84^\circ = 96^\circ$

Step7: Relate central angle to arc

The measure of arc $\overset{\frown}{QP}$ equals the measure of its central angle $\angle QTP$.

Answer:

96°