Definition 6

A segment of a circle is the figure contained by a straight line and a circumference of a circle.

Definition 7

An angle of a segment is that contained by a straight line and a circumference of a circle.

Definition 8

An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the ends of the straight line which is the base of the segment, is contained by the straight lines so joined.

Definition 9

And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference.

Guide

A line in a circle, such as the line BC, divides the circle into two segments, the small blue segment BDC, and the large yellow segment BEC.

An angle of the segment BDC is not a rectilinear angle, since only one of its sides, BC, is a straight line. The other side is curved, namely, an arc of a circle. These angles of segments only appear in proposition III.16, and are not important in Euclid’s development of geometry.

An example of an angle in a segment is the angle BFC in the yellow segment BEC. This angle BFC stands upon the circumference (arc) BDC. Angles in segments are rectilinear, and they are important. In proposition III.21, Euclid proves that all the angles in a given segment are equal.