The internal bisectors of $\angle Q $ and $\angle R $ of $\Delta PQR $ meet at $ S $. If $ m$\angle P$ = 75^\c

Question

The internal bisectors of $\angle Q $ and $\angle R $ of $\Delta PQR $ meet at $ S $. If $ m$\angle P$ = 75^\circ $, then $ m$\angle QSR$ $ is:

Examoogle · Accepted Answer

The correct answer is Option D: $127.5^\circ$. Remember that the angle between the internal bisectors of two interior angles of a triangle is given directly by $90^{\circ} + \frac{	ext{third angle}}{2}$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Remember that the angle between the internal bisectors of two interior angles of a triangle is given directly by $90^{\circ} + \frac{	ext{third angle}}{2}$.

Formula Used: $$\angle QSR = 90^{\circ} + \frac{1}{2}\angle P$$

Given: m\angle P = 75^{\circ}

Steps: Understand the Geometry Property: In any triangle $	riangle PQR$, if the internal bisectors of $\angle Q$ and $\angle R$ meet at point $S$, the angle $\angle QSR$ is given by the standard formula involving the opposite vertex angle $\angle P$.. Substitute the Given Value: We are given that $	ext{m}\angle P = 75^{\circ}$. Substitute this value into the formula.. Calculate the Result: Compute $\frac{75}{2} = 37.5^{\circ}$, then add it to $90^{\circ}$ to find the final angle measure.

Answer: B is correct because the angle formed by the internal bisectors of \u2220Q and \u2220R is 90\u00b0 + (75\u00b0 / 2) = 102.5\u00b0.

Exam Tip: This property of incenter angle is frequently used in geometry questions involving circumcircles, incenters, and polygon interior angles.

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