Question : $\triangle \mathrm{PQR}$ is an equilateral triangle inscribed in a circle. $\mathrm{S}$ is any point on the arc $\mathrm{QR}$. Find the measure of $\angle \mathrm{PSQ}$.
Option 1: $30^{\circ}$
Option 2: $60^{\circ}$
Option 3: $90^{\circ}$
Option 4: $45^{\circ}$
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Correct Answer: $60^{\circ}$
Solution : Since $\triangle PQR$ is an equilateral triangle, $\angle PQR = \angle QRP = \angle RPQ = 60°$ Angles subtended by a chord on the same side of a circle are equal. Here, $\angle QRP$ and $\angle PSQ$ are angles subtended on the circle by chord $PQ$. So, $\angle PSQ = \angle QRP = 60°$ Hence, the correct answer is $60^{\circ}$.
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Question : If $\triangle \mathrm{ABC} \cong \triangle \mathrm{PQR}, \mathrm{BC}=6 \mathrm{~cm}$, and $\angle \mathrm{A}=75^{\circ}$, then which one of the following is true?
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Question : In a $\triangle ABC$, if $\angle A=90^{\circ}, AC=5 \mathrm{~cm}, BC=9 \mathrm{~cm}$ and in $\triangle PQR, \angle P=90^{\circ}, PR=3 \mathrm{~cm}, QR=8$ $\mathrm{cm}$, then:
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Question : In $\triangle P Q R, S$ is a point on the side QR such that $\angle Q P S=\frac{1}{2} \angle P S R, \angle Q P R=78^{\circ}$ and $\angle P R S=44^{\circ}$. What is the measure of $\angle PSQ$?
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