Question

In the given figure O is the centre of the circle if angle p o r is equal to 140 degree find the angle P Q R and PSR

Answer 1

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Answer 2

The value of angles are $\red{\angle PQR=70^{\circ}}\: and\:\red{\angle PSR=110^{\circ}}$

Given:

• A figure.
• O is the centre of the circle if $\angle POR=140^{\circ}\:$.

To find:

• Find the $\angle PQR\:and\:\angle PSR$

Solution:

Theorem/Concept to be used:

1. Angle subtended by an arc/chord on centre of circle is double of angle subtended by that arc on segment of circle.
2. Sum of opposite angles of cyclic quadrilateral is 180°.

Step 1:

As the chord PR subtends angle $\angle POR=140^{\circ}\:$.

and the same chord subtends $\angle PQR\:$.

Thus,

According to the theorem 1,

$\angle PQR=\frac{1}{2}\angle POR\\\:$

or

$\bf \red{\angle PQR=70^{\circ}}$

Step 2:

PQRS is a cyclic quadrilateral, as all its vertices are lying on the perimeter of circle.

Thus,

$\angle PQR+\angle PSR=180^{\circ}$

or

$70^{\circ}+\angle PSR=180^{\circ}$

or

$\angle PSR=180^{\circ}-70^{\circ}$

or

$\bf \red{\angle PSR=110^{\circ}}$

Thus,

The value of angles are $\angle PQR=70^{\circ}\: and\:\angle PSR=110^{\circ}$

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