Two Straight Lines make Equal Opposite Angles

Theorem
If two straight lines cut each other, they make the opposite angles equal each other.

Porism
It follows that if two straight lines cut one another, the angles at the point of intersection make four right 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 $$\angle BEC = \angle AED$$.

Note
This is Proposition 15 of Book I of Euclid's "The Elements".