OP, OQ, OR and OS are concurrent. OP and OR are on a same straight line. P and R are situated on opposite sides of the point O.

∠POQ = ∠ROS and ∠POQ = ∠QOR If ∠POQ = 50° then write the measurement of ∠QOR, ∠ROS and ∠POS.

Now let us first construct a diagram from the explanation given in the question.

It is mentioned that OP, OQ, OR and OS are concurrent.

Concurrent means multiple line intersecting at a single common point.

If we carefully look at these rays, each ray has common point O. So, the lines intersect at a common point O.

Now it is given that OP and OR is on same straight line. At the same time P and R are situated on opposite sides of point O.

So, our first straight line is PR which intersecting with line SQ at a common point O.

In the above given figure, let us understand the pairs of vertically opposite angle.

∠POQ and ∠SOR are lying opposite to each other [in common language lying face to face to each other].

So, first pair of vertically opposite angles is ∠POQ and ∠SOR.

∠POS and ∠QOR are also lying opposite to each other.

Hence ∠POS and ∠QOR forms our second pair of vertically opposite angles.

Let ∠POQ = ∠1

∠QOR = ∠2

∠SOR = ∠3

∠POS = ∠4.

Vertically opposite angles are always equal.

Hence ∠1 must be equal to ∠3 and ∠4 has to be equal to ∠2.

Also, we know that sum all vertically opposite angles is 360°.

∠1 + ∠2 + ∠3 + ∠4 = 360°.

Now in the question it is given that,

∠1 = 50°

So ∠3 must be equal to 50° [as ∠1 = ∠3……… vertically opposite angles]

Let ∠4 = ∠2 = x [as ∠2 = ∠4……… vertically opposite angles]

So, 50° + 50° + x + x = 360°

100° + 2x = 360°

2x = 360 – 100

2x = 260

x = 260 / 2

x = 130°

So ∠2 = ∠4 = 130°.

So ∠POQ = ∠SOR = 50°.

∠POS = ∠QOR = 130°.