Question

In the given fig POQ is the diameter of the circle and <POR =110°. Find < QSR.​

Answer 1

Given :-

• A C(O, r) such that POQ is a diameter and ∠POR = 110°

To Find :-

• Value of ∠QSR

Concept Used :-

1. Linear Pair :- Linear pair of angles are formed when two lines intersect each other at a single point. The angles are said to be linear if they are adjacent to each other after the intersection of the two lines. The sum of angles of a linear pair is always equal to 180°

2. The angle subtended by an arc at the centre is twice the angle subtended at the circumference.

Solution :-

Given that,

• ∠POR = 180°.

As POQ is a line.

• ⇛ ∠POR + ∠QOR = 180°

⇛ 110° + ∠QOR = 180°

• ⇛ ∠QOR = 180° - 110° = 70°.

Now,

We know, The angle subtended by an arc at the centre is twice the angle subtended at the circumference 

Since, ∠QOR is subtened at the centre O and ∠QSR is subtended on the circumference of a circle by same arc RQ.

Therefore,

• ⇛ ∠ROQ = 2 × ∠QSR

• ⇛ 2 × ∠QSR = 70°

• ⇛ ∠QSR = 35°

ADDITIONAL INFORMATION :-

Two equal chords of a circle subtend equal angles at the centre of the circle.

If two angles subtended at the center by two chords are equal, then the chords are of equal length.

The perpendicular to a chord bisects the chord if drawn from the centre of the circle.

A straight line passing through the centre of a circle to bisect a chord is perpendicular to the chord.

Equal chords of a circle are equidistant (equal distance) from the centre of the circle.

The opposite angles in a cyclic quadrilateral are supplementary

Angle in same segments are equal.