Question : In the figure, in $\triangle {PQR}, {PT} \perp {QR}$ at ${T}$ and ${PS}$ is the bisector of $\angle {QPR}$. If $\angle {PQR}=78^{\circ}$, and $\angle {TPS}=24^{\circ}$, then the measure of $\angle {PRQ}$ is:
Option 1: 42$^{\circ}$
Option 2: 39$^{\circ}$
Option 3: 30$^{\circ}$
Option 4: 40$^{\circ}$
Correct Answer: 30$^{\circ}$
Solution : In $\triangle$PQR, PT $ \perp$ QR at T and PS is the bisector of $\angle$QPR $\angle$PQR = 78° and $\angle$TPS = 24° In $\triangle$PQT, $\angle$QPT = 180$^\circ$ – $\angle$PQR – $\angle$PTQ = 180$^\circ$ – 78$^\circ$ – 90$^\circ$ = 12$^\circ$ $\angle$SPQ = $\angle$QPT + $\angle$TPS = 12$^\circ$ + 24$^\circ$ = 36$^\circ$ Since PS is the bisector of $\angle$ QPR, $\angle$QPR = 2 × $\angle$SPQ = 2 × 36$^\circ$ = 72$^\circ$ In $\triangle$PQR, $\angle$ PRQ = 180$^\circ$ – $\angle$ PQR – $\angle$ QPR = 180$^\circ$ – 78$^\circ$ – 72$^\circ$ = 30$^\circ$ Hence, the correct answer is 30$^\circ$.
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