COROLLARY I. An angle standing upon a semicircumference is a right angle.

COROLLARY II. An angle standing upon a circular arc is an acute angle, and an angle standing upon a semicircular arc is an obtuse angle.

COROLLARY III. Angles standing upon opposite arcs are supplementary to each other.

COROLLARY IV. The circumference described on a straight line which subtends a right angle, passes through the vertex.

THEOREM 34.

The measure of an angle the vertex of which is within the circumference, is half its arc together with half that of its vertical.

Let A B C be an angle the vertex of which is within the circumference.

Let its sides be produced beyond the vertex to D and E, and let E H be a parallel to A D. Because the corresponding angles A B C and H E C are equal to each other [20 i], and half the arc HC is the measure of the angle HEC [34], half the arc HC is also the measure of the angle ABC [i]. But the arc HC is

E

D

composed of the arcs A C and H A, and the arc HA is equal to the arc ED [I. 53]; therefore half the arc A C together with half the arc E D form the measure of the angle AB C, which W. T. B. D. COROLLARY I. If an angle be formed by two secants, and its vertex be

without the circumference, its measure is half the difference of the arcs intercepted between its sides.

COROLLARY II. If the vertices of two supplementary angles subtended by the same straight line, be on opposite sides of the latter, the vertices and the extremities of the subtending line are points of the same circumference.

COROLLARY III. If two chords of a circumference cut each other perpendicularly, the arcs of two vertical angles thus formed, are together equal to a semicircumference.

THEOREM 35.

The measure of an angle formed by a tangent and a chord terminating at the point of tangence, is half the arc subtended by the chord.

Let ABC be an angle formed by a tangent A B to the circumference BCD, and the chord BC terminating at the point of tangence B.

D

B

Let CD be a chord parallel to AB. Because the alternate angles ABC and B C D are equal to each other [20], and half the arc BD is the

measure of the angle BCD [33], half the arc BD is also the measure of the angle ABC. But the arcs B C and B D are equal to cach other [I. 55 iii]; therefore, half the arc BC is the measure of the angle ABC, which W. T. B. D.

COROLLARY. Conversely, if the measure of the angle formed by a chord and a straight line passing through its extremity be equal to half the arc subtended by the chord, the straight line is tangent to the circumference.

Scholium. The measure of the angle formed by a tangent and the diameter terminating at the point of tangence, is the fourth part of the circumference (I. 55).

THEOREM 36.

The measure of a circumscribed angle is half the difference of its ares.

Let A B C be an angle formed by the two tangents BA and B C, and let C and D be the points of tangence.

C

H

Let DE be a chord parallel to B C through the point D. Because the measure of the angle A D E is half the arc D E [35], and the cor

responding angles ADE and A B C are equal to each other [20], the measure of the angle ABC is also half the arc DE [i]. But the arcs E C and D C are equal to each other [I. 53]; the arc D E is then equal to the difference of the arcs CED and CHD; therefore the measure of the angle ABC is half the difference of the arcs CED and CHD, which W. T. B. D.

COROLLARY. The measure of an angle formed by a tangent and a secant, the vertex of which is without the circumference, is half the difference of the arcs intercepted by its sides.

Scholium. The angle formed by two straight lines tangent at the extremities of a diameter is nought (I. 55 ii).

CHAPTER II.

ON PROPORTIONAL LINES.

SECTION I.

ON STRAIGHT LINES CUT BY PARALLELS.

DEFINITIONS.

19. Straight lines are proportionally, or similarly, divided when any two adjoining parts of one of them are proportional to two adjoining parts of each of the others; the points of division separating the homologous parts are called homologous points.

20. When the sides of an angle are cut proportionally by straight lines, the portions of the latter intercepted between the sides, subtend the angle proportionally.

THEOREM 37.

If two parallels which cut both sides of an angle, determine on one side two equal portions thereof, the portions of the other side are equal to each other.

Let A B C be an angle the sides of which are cut by the parallels DE and A C, and let the portions BD and D A determined by the parallels on the side B A, be equal to each other.

Let B H be a parallel to D E and AC [I. 17 iii], through the point