Angles in Same Segment of Circle are Equal

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In the words of Euclid:

In a circle the angles in the same segment are equal to one another.

(The Elements: Book $\text{III}$: Proposition $21$)



Let $ABCD$ be a circle, and let $\angle BAD, \angle BED$ be angles in the same segment $BAED$.

Let $F$ be the center of $ABCD$, and join $BF$ and $FD$.

From the Inscribed Angle Theorem:

$\angle BFD = 2 \angle BAD$
$\angle BFD = 2 \angle BED$

So $\angle BAD = \angle BED$.

Hence the result.


Historical Note

This theorem is Proposition $21$ of Book $\text{III}$ of Euclid's The Elements.