Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles/Porism

Theorem

In the words of Euclid:

From this it is manifest that, if there be two equal plane angles, and if there be set up on them elevated straight lines which are equal and contain equal angles with the original straight lines respectively, the perpendiculars drawn from their extremities to the planes in which are the original angles are equal to one another.

(The Elements: Book $\text{XI}$: Proposition $35$ : Porism)

Proof

Directly apparent from the construction.

$\blacksquare$

Historical Note

This proof is Proposition $35$ of Book $\text{XI}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Propositions