Straight Lines which make Equal Angles with Perpendicular to Straight Line are Equal

Theorem
Let $AB$ be a straight line.

Let $C$ be a point which is not on $AB$.

Let $D$ be a point on $AB$ such that $CD$ is perpendicular to $AB$.

Let $E, F$ be points on $AB$ such that $\angle DCE = \angle DCF$.

Then $CE = CF$.

Proof
$\triangle CDE$ and $\triangle CDF$ are right triangle where $CE$ and $CF$ are the hypotenuses.

We have:
 * $\angle CDE = \angle CDF$ as both are right angles.


 * $\angle DCE = \angle DCF$ by hypothesis.


 * $CD$ is common.

Thus by Triangle Angle-Side-Angle Equality, $\triangle CDE$ and $\triangle CDF$ are congruent.

Hence the result.