Question

Find angle measure of PQR. Explain your reasoning

Answer 1

Answer:

Answer: The required measure are m∠PQR = 60° and m∠PSR = 120°.

Step-by-step explanation: Given that in the figure shown, O is the center of the circle and m∠POR=120°.

We are to find the measures of ∠PQR and ∠PSR.

From the figure, we note that

PQRS is a cyclic quadrilateral.

We know that

the measure of the angle subtended by an arc at the center of the circle is double of the measure of angle subtended by the same arc on the circumference.

So, in the given circle, on arc PR, we must have

\begin{gathered}m\angle POR=2\times m\angle PQR\\\\\Rightarrow m\angle PQR=\dfrac{m\angle POR}{2}=\dfrac{120^\circ}{2}=60^\circ.\end{gathered}

m∠POR=2×m∠PQR

⇒m∠PQR=

2

m∠POR

=

2

120

∘

=60

∘

.

Also, the sum of the opposite angles of a cyclic quadrilateral is 180° .

Therefore, we get

\begin{gathered}m\angle PQR+m\angle PSR=180^\circ\\\\\Rightarrow m\angle PSR=180^\circ-m\anglePQR=180^\circ-60^\circ=120^\circ.\end{gathered}

m∠PQR+m∠PSR=180

∘

⇒m∠PSR=180

∘

−m\anglePQR=180

∘

−60

∘

=120

∘

.

Thus, the required measure are m∠PQR = 60° and m∠PSR = 120°.