When two lines intersect, they create pairs of angles known as vertically opposite angles, which are always equal.

To illustrate this, consider two straight lines crossing each other, forming four angles labeled as $A$, $B$, $C$, and $D$. In this configuration, angle $A$ is vertically opposite to angle $C$, and angle $B$ is vertically opposite to angle $D$. A key property of vertically opposite angles is their equality: if angle $A$ measures $50^\circ$, then angle $C$ will also measure $50^\circ$, and this holds true for angles $B$ and $D$ as well.

This property of vertically opposite angles is extremely useful in solving various geometric problems. For example, if you know the measure of one angle formed by the intersection of two lines, you can easily determine the measure of its vertically opposite angle. This simplicity can facilitate calculations and assist in finding unknown angles in more complex geometric figures.

To understand why vertically opposite angles are equal, consider that the sum of the angles around a point is $360^\circ$. When two lines intersect, they form four angles, and the angles along a straight line sum up to $180^\circ$. Consequently, the two pairs of opposite angles must each add up to $180^\circ$. Since each pair of opposite angles shares this property, they must be equal to one another.

In summary, the equality of vertically opposite angles is a fundamental concept in geometry that aids in the analysis and resolution of problems involving intersecting lines.