Vertically Opposite Angles: Proving the Equality Theorem

To solve this problem, we will explore the concept of vertically opposite angles:

When two straight lines intersect each other at a point, they form two pairs of opposite angles. These pairs of angles are called vertically opposite angles.

The theorem of vertically opposite angles states that they are always equal to each other. Here's why:

• Consider two intersecting lines, forming four angles at their intersection point.
• These angles are labeled $\angle 1$, $\angle 2$, $\angle 3$, and $\angle 4$.
• The angles $\angle 1$ and $\angle 3$ are vertically opposite, as are $\angle 2$ and $\angle 4$.
• Since linear pairs of angles are supplementary, we have:
• $\angle 1 + \angle 2 = 180^\circ$
• $\angle 2 + \angle 3 = 180^\circ$
• From these equations, by setting the expressions for $\angle 2$ equal, we obtain that $\angle 1 = \angle 3$.
• Similarly, $\angle 2 = \angle 4$ by using the same reasoning.

Thus, it is established that vertically opposite angles are indeed equal.

Therefore, the statement "vertically opposite angles are equal to each other" is True.