Sum of measurement of ∠POR and ∠QOS is 110^o. Let’s write the measurement of ∠POS, ∠QOS, ∠QOR and ∠POR.

Let us first find out the pairs of vertically opposite angles from the above given figure.

Now from the above figure,

∠POS is lying opposite to ∠ROQ

So, our first pair of vertically opposite angles are ∠POS and ∠ROQ.

Now when we again observe the figure, we notice that ∠POR and ∠SOQ are also lying opposite to each other.

So, our second pair of vertically opposite angles is ∠POR and ∠SOQ.

Now we know two basic properties of vertically opposite angles which are as follows:

1. Vertically opposite angles are always equal.

So, we can say that ∠POS = ∠ROQ and ∠POR = ∠SOQ

2. Sum of all vertically opposite angles is 360°.

∠POS + ∠ROQ + ∠POR + ∠SOQ = 360° ……... (i)

Now in the question it given that sum of angles ∠POR and ∠SOQ is 110°.

Let ∠POR = ∠SOQ = x [as these both angles are equal]

So, x + x = 110°

2x = 110°

x = 110 / 2

x = 55°

So ∠POR = ∠SOQ = 55°.

Let ∠POS = ∠ROQ = x [as these both angles are equal]

Let us substitute the values in equation (i)

55° + 55° + x + x = 360°

110° + 2x = 360°

2x = 360 – 110

2x = 250

x = 250 / 2

x = 125°

∠POS = ∠ROQ = 125°.