### Theory:

Let us discuss what vertically opposite angles are.
Definition:
When two lines intersect at a particular point, the vertically opposite angles formed are equal in measure.
Illustration: In the figure, the lines $$p$$ and $$q$$ intersect at $$O$$ forming four pair of angles $$a$$, $$b$$, $$c$$ and $$d$$.

The angle $$a$$ is vertically opposite to $$c$$, and the angle $$b$$ is vertically opposite to $$d$$ and vice versa.

Here $$\angle a$$ $$=$$ $$\angle c$$ and $$\angle b$$ $$=$$ $$\angle d$$.

Let us prove the above equivalence as follows:

Consider $$\angle a$$ and $$\angle b$$.

These two angles form a linear pair. Hence by the property of linear pair of angles $$\angle a + \angle b = 180^{\circ}$$.

$$\Rightarrow$$ $$\angle a$$ $$=$$ $$180^{\circ}$$ $$-$$ $$\angle b$$          ……$$(1)$$

Also, $$\angle b$$ $$=$$ $$180^{\circ}$$ $$-$$ $$\angle a$$      ……$$(2)$$

Consider $$\angle b$$ and $$\angle c$$.

These two angles form a linear pair. Hence by the property of linear pair of angles $$\angle b + \angle c = 180^{\circ}$$.

$$\Rightarrow$$ $$\angle c$$ $$=$$ $$180^{\circ}$$ $$-$$ $$\angle b$$          ……$$(3)$$

Thus from equations $$(1)$$ and $$(3)$$ we have:

$$\angle a$$ $$=$$ $$\angle c$$

Similarly, consider $$\angle a$$ and $$\angle d$$.

These two angles form a linear pair. Hence by the property of linear pair of angles $$\angle a + \angle d = 180^{\circ}$$.

$$\Rightarrow$$ $$\angle d$$ $$=$$ $$180^{\circ}$$ $$-$$ $$\angle a$$          ……$$(4)$$

And from equations $$(2)$$ and $$(4)$$ we have:

$$\angle b$$ $$=$$ $$\angle d$$

Therefore, the vertically opposite angles formed by the lines $$p$$ and $$q$$ at the point $$O$$ are equal in measure.
Example:
Find the unknown angle $$b$$ in the figure. Solution:

From the figure, we observe that the $$\angle SOU$$ and $$\angle TOV$$ are vertically opposite angles formed by the line segments $$ST$$ and $$UV$$.

By definition, the vertically opposite angles are equal.

Thus $$\angle SOU$$ $$=$$ $$\angle TOV$$.

$$\Rightarrow$$ $$b$$ $$=$$ $$50^{\circ}$$

Therefore, the unknown angle $$b$$ is $$50^{\circ}$$.