Prop. 27. (converse to Prop. 26). In equal circles the angles which stand on equal arcs are equal to one another, whether they be at the centres or the circumferences;
Prop. 28. In equal circles equal straight lines (equal chords) cut off equal arcs, the greater equal to the greater, and the less equal to the less;
Prop. 29 (converse to Prop. 28). In equal circles equal arcs are subtended by equal straight lines.
§ 35. Other important consequences of Props. 20-22 are:—
Prop. 31. In a circle the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle;
Prop. 32. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.
§ 36. Propositions 30, 33, 34, contain problems which are solved by aid of the propositions preceding them:—
Prop. 30. To bisect a given arc, that is, to divide it into two equal parts;
Prop. 33. On a given straight line to describe a segment of a circle containing an angle equal to a given rectilineal angle;
Prop. 34. From a given circle to cut off a segment containing an angle equal to a given rectilineal angle.