Two Straight Lines make Equal Opposite Angles

Proof


Let $AB$ and $CD$ be two straight lines that cut each other at the point $E$.

Since the straight line $AE$ stands on the straight line $CD$, the angles $\angle AED$ and $\angle AEC$ make two right angles.

Since the straight line $DE$ stands on the straight line $AB$, the angles $\angle AED$ and $\angle BED$ make two right angles.

But $\angle AED$ and $\angle AEC$ also make two right angles.

So by Common Notion 1 and the fact that all right angles are congruent, $\angle AED + \angle AEC = \angle AED + \angle BED$.

Let $\angle AED$ be subtracted from each.

Then by Common Notion 3 it follows that $\angle AEC = \angle BED$.

Similarly it can be shown that $\angle BEC = \angle AED$.

Also known as
This result is also called the vertical angle theorem.

The arises from the fact that the angles proven equal are known as vertical angles.