### Theory:

Theorem $$1$$: When two lines intersect, then the vertically opposite angles are equal.

Proof:

Let $$AB$$ and $$BD$$ be two line segments intersecting at $$O$$ as given in the figure.

We should prove that the vertically opposite angles are equal.

The vertically opposite angles are:

1. $$\angle AOD$$ and $$\angle BOC$$

2. $$\angle AOC$$ and $$\angle BOD$$

Let us consider the vertically opposite angles $$\angle AOD$$ and $$\angle BOC$$, and prove that they are equal.

$$OD$$ is a ray standing on the line $$AB$$.

$$\angle AOD + \angle BOD = 180^\circ \longrightarrow (1)$$

[By linear pair of angles axiom 1]

Similarly, $$\angle BOD + \angle BOC = 180^\circ \longrightarrow (2)$$

Let us now equate $$(1)$$ and $$(2)$$.

$$\angle AOD + \angle BOD = \angle BOD + \angle BOC$$

Thus, $$\angle AOD = \angle BOC$$

Hence, the vertically opposite angles formed by two intersecting lines are equal.