Vertical angles can be defined as the pairs of non-adjacent angles in opposite positions formed by the intersection of two straight lines are known as vertical angles.
Vertical Angles
In the figure, $\angle$1 and $\angle$3 are a pair of vertical angles. Another pair of vertical angles is $\angle$2 and $\angle$4.

If two angles are vertical angles, then they have equal measures.

Construction: Let $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ be two intersecting lines in a plane, intersecting at the point P, as shown in the figure. Here, the angles $\angle$APD and $\angle$BPC are vertical angles.
Vertical Angles Theorem

To Prove:
$$\angle APD \cong \angle BPC$$
Proof:
S. No.StatementReason
1∠APD+∠APC = 180oAngles on a straight line (Supplementary angles)
2∠BPC+∠APC = 180oAngles on a straight line
3∠APD+∠APC=∠BPC+∠APCFrom (1) & (2)
4∠APC≅∠BPC(3), Subtracted ∠APC from both sides

Hence, we proved the theorem.