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

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

I say that the angles *BAD* and *BED* equal one another.

Take the center *F* of the circle *ABCD,* and join *BF* and *FD.*

Now, since the angle *BFD* is at the center, and the angle *BAD* at the circumference, and they have the same circumference *BCD* as base, therefore the angle *BFD* is double the angle *BAD.*

For the same reason the angle *BFD* is also double the angle *BED.* Therefore the angle *BAD* equals the angle *BED.*

Therefore *in a circle the angles in the same segment equal one another.*

Q.E.D.