to CD [I. 29], and will coincide with each other; hence the arc CA will coincide with the arc CB, and the radius OA with the radius O B: that is, the sector A O C M will coincide with the sector BOCN; therefore they are equal to each other, W. W. T. B. D.

COROLLARY I. The diameter perpendicular on the chord of the arc of a sector, bisects the angle of the sector.

COROLLARY II. The diameter perpendicular on the base of a segment bisects the segment.

COROLLARY III. A segment is bisected by its altitude.

THEOREM 74.

Equal segments are parts of equal circles.

Let ABM and DEN be two equal segments.

Let A B M be a portion of the circle O and DEN a portion of the circle C, and let the two segments be applied to each other. Because the segment A B M is equal to the segment DEN, they will coincide with each other [Gen. Def. 18], and the arc A M B will coincide with the arc DNE; hence the centres O and C will coincide with each other [I. 46 i], and the circumference O will coincide with the circumference C [I. 58]: therefore the circle O is equal to the circle C, which W. T. B. D. Inversely

Unequal circles cannot have equal segments.

COROLLARY I. Equal sectors are parts of equal circles.

COROLLARY II. Equal segments are parts of equal sectors, and conversely.

THEOREM 75.

Two semicircular segments of a circle, which have equal bases, are equal to each other.

Let A B M and CDN be two segments of the circle ABC, having the base A B equal to the base C D.

Let H be the middle point of the arc A C; let HI be a diameter through the point H, and let the circle ABC be folded upon HI: then the semicircle HIM will coincide with the semicircle HIN [72]. Because the arc HA is equal to the arc HC [Const.], the point A will coincide with the point

C; because the chord A B is equal to the chord C D, the arc A M B is equal to the arc

CND [I. 43]: hence the arc A M B will coincide with the arc CND, and the chord AB with the chord CD; that is, the segment ABM will coincide with the segment CDN; therefore they are equal to each other, W. W. T. B. D. Inversely

Two unequal segments of a circle have unequal bases, and

Unequal semicircular segments which have equal bases, are not portions of the same circle.

COROLLARY I. Two circular segments of a circle, which have equal bases, are equal to each other.

COROLLARY II. Segments of equal circles which have equal bases, are equal to one another.

COROLLARY III. Segments of a circle, or of equal circles, which have equal arcs, are equal to one another.

THEOREM 76.

Sectors of a circle which have equal angles are equal to each other.

Let A OBM and COD N be two sectors of the circle O, having the angle A OB equal to the angle C O D.

Let the diameter H I

bisect the angle A O C, and let the circle O be folded upon HI; then the semicircle HIM will coincide with the semicircle HIN [72]. Because the angle HOA is equal to the angle HOC [Const.], the radius OA will coincide with the radius O C; because the angle AO B is equal to the angle COD, the radius OB will coincide with the ra

dius OD: then the sector

AO B M will coincide with the sector CODN; therefore they are equal to each other, W. W. T. B. D. Inversely

Unequal sectors of a circle have unequal angles, and

Unequal sectors having equal angles are not portions of the same circle. COROLLARY I. Sectors of equal circles which have equal angles are equal to one another.

COROLLARY II. Sectors of a circle, or of equal circles, which have equal arcs, are equal to one another.

A

THEOREM 77.

(Eucl. III. 31.)

The angle inscribed in a semicircle is right.

Let A C B be an angle inscribed in one half of the circle O.

Let ABD be the other half of the circle O; then the angle AC B is inscribed in the circumference O, and stands upon the arc AD B. Because ABD is a semicircle, the base AB is a diameter [72 i]; then the arc ADB is half the circumference ACD [I.39]; therefore the inscribed angle ACB is a right angle [II. 33 i], W. W. T. B. D.

COROLLARY I. Conversely, a segment capable of a right angle, is a semicircle.

COROLLARY II. A greater segment is capable of a smaller angle, and conversely (II. 27 iii).

Scholium. A semicircular segment is capable of an obtuse angle, and a circular segment is capable of an acute angle (II. 33 ii).

THEOREM 78.

The angle of a sector is equal to twice the supplement of the angle inscribed in its segment.

Let A OBC be a sector of the circle O; let A B C be its segment, and let A CB be an angle inscribed in that segment.

Let the angle A D B be inscribed in the segment ABD; then the angles AC B and AD B are supplementary to each other [II. 33 iii]. Because the centre angle AO B and the inscribed angle ADB stand.

upon the same arc, the angle ADB is equal to half the angle

AOB [II. 27]; therefore the

angle AOB is equal to twice the supplement of the angle AC B, which W. T. B. D.

COROLLARY I. The angle inscribed in a segment is equal to half the explement of the angle of its sector.

COROLLARY II. Any segment is capable of the supplement of the angle inscriptible in the opposite segment (II. 33 iii).

COROLLARY III. All the angles

D

C

B

inscribed in a segment, or in equal segments, are equal to one another. COROLLARY IV. If two sectors have equal angles, the angles inscribed in their segments are equal one to another, and conversely.

COROLLARY V. The angle formed on one side of a chord by a tangent to the circle, is inscriptible in the opposite segment formed by that chord (II. 29 ii).

THEOREM 79.

Two segments of a circle which are capable of the same angle, are equal to each other.

Let A B C and D E F be two segments of the circle O, capacapable of the same angle.

Let the points A and B be joined to the centre and to a point C of the arc ACB; let also the points D and F be joined to the centre and to a point F of the arc DFE: then the angles ACB and D FE

are inscribed in the segments

E

B