Two Straight Lines make Equal Opposite Angles
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
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 proof is Proposition $15$ of Book $\text{I}$ of Euclid's The Elements.
It appears to have originally been created by Thales of Miletus.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions
- 1968: M.N. Aref and William Wernick: Problems & Solutions in Euclidean Geometry ... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.3$
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.1$: Thales (ca. $\text {625}$ – $\text {547}$ B.C.)