Find the angle between two chords passing through points where lines are tangent to the circle

Question

In the given figure, PQ and PR are tangents to the circle with centre $O$ and $S$ is a point on the circle such that $\angle{SQL}={50}^{\circ}$ and $\angle{SRM}={60}^{\circ}$. Find $\angle{QSR}.$

What I've tried,

Join $OQ$ and $OR$. Since the line joining the point of contact of the tangent to the centre of the circle is equal to $90^{\circ}$.

$\therefore$$\angle$OQL=$\angle{ORL}={90}^{\circ}$

But, now I am stuck.

Answer 1

Draw your two radii. $\triangle SOR$ is isoceles and $\angle SRO=30^\circ=\angle RSO$. Similarly $\angle SQO=40^\circ=\angle QSO$, so $\angle QSR=70^\circ$

Answer 2

Since $\angle QRS = \angle LQS = 50$ and $\angle RQS = \angle SRM = 60$, we have $\angle QSR = 180-\angle QRS-\angle RQS = 180-50-60=\boxed{70^\circ}$

Answer 3

The big tools you need with this specific problem is that:

1) all radiuses are equal

2) isosceles triangle theorem (angles opposite... are equal)

3) Tangents to the circle make a 90 degree angle with the segment from point of intersection and the center of the circle.

Take note that

$$OS=OR$$

$$OS=OQ$$

Draw the tangents and add the angles $30$ and $40$ which I see you've erased not so throughly. Also take note of any isosceles triangles. Then I hope the problem will be obvious to you.