Two Straight Lines make Equal Opposite Angles

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Theorem

In the words of Euclid:

If two straight lines cut one another, they make the vertical angles equal to one another.

(The Elements: Book $\text{I}$: Proposition $15$)


Porism

It follows that if two straight lines cut one another, the angles at the point of intersection make four right angles.


Proof

Euclid-I-15.png

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 $\angle BEC = \angle AED$.

$\blacksquare$


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.


Historical Note

This theorem is Proposition $15$ of Book $\text{I}$ of Euclid's The Elements.
It appears to have originally been created by Thales of Miletus.


Sources