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139. In the same circle, or equal circles, equal angles at the center intercept equal arcs on the circumference.

Let O and O', Fig. 87, be equal circles, and AOB and A' O'B' equal angles. Place the circle O' on

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O so that the point O' coincides with O and the line O'B' takes the direction O B. Then, since O B and O'B' are equal, being radii of equal circles, B' will fall on B, and, since the angle O' is equal to the angle O, the line O' A' will take the direction of OA, and, being equal to OA, its extremity A'

will fall on A. Hence, the arcs AB and A'B' will coincide and are equal.

140. In the same circle, or equal circles, equal arcs are intercepted by equal angles at the center.

Let O and O', Fig. 87, be equal circles, and A B and A' B' equal arcs. Place the circle O' on the circle O, with the points O' and A' on O and A, respectively. Then, since the arc A'B' is equal to the arc AB, B' will fall on B. Then the angle O' is equal to the angle O, as the vertex and the sides of the angles coincide.

141. In the same circle or equal circles, equal chords. subtend equal arcs.

Let A B and CD, Fig. 88, be equal chords. Draw the radii A O, B O, CO, and DO, joining A, B, C and D to 0. Then the triangles AOB and COD, having three sides of one equal to three sides of the other, are equal. Hence, the angle AOB is equal to the angle COD, and, therefore (Art. 139), the arc A B is equal to the arc CD.

FIG. 88

142. In the same circle, or equal circles, equal arcs are subtended by equal chords.

143. A perpendicular from the center of a circle to a chord bisects the chord and the arc subtended by it.

Let OM, Fig. 89, be drawn from O perpendicular to the chord AB. Join O to A and B. The triangle AOB is isosceles, since the two sides OA and OB are radii of the same circle. Therefore (Art. 77), AM = MB. Also, A OM = MOB (Art. 77); therefore (Art. 139), arc AC arc CB.

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144. The perpendicular erected at the middle of a chord passes through the center of the circle and bisects the arc subtended by the chord.

145. Through any three points not in a straight line a circumference can be passed.

H

FIG. 90

B

Let A, B, and C, Fig. 90, be any three points. Draw A B and B C. At the middle point of A B 4 draw KH perpendicular to A B; at the middle point of CB draw FE perpendicular to B C and meeting KH at 0. As O is a point in the perpendiculars at the middle points of AB and BC, it is equally distant from A, B, and C. Therefore, a circle with O as center and OB as radius will pass through A, B, and C.

146. A straight line perpendicular to a radius at its extremity is tangent to the circle.

Let A B, Fig. 91, be perpendicular to OH at its extremity H. As OH is perpendicular to AB it is shorter than any other line, as OM, drawn from 0 to AB. Hence, M is without the circle, and any point in AB other than H is without the circle. Therefore, AB touches the circle in only the point H, and is, consequently, tangent to the circle.

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147. A perpendicular to a tangent at the point of tangency passes through the center of the circle.

148. A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

149. If two circles intersect, the line joining their centers bisects at right angles the line joining the points where the circles intersect.

B

FIG. 92

-P

Let the two circles whose centers are O and P, Fig. 92, intersect at A and B. The point P, being the center of a circle, is equally distant from A and B, points on the circumference. Similarly, O is equally distant from A and B. Hence, by Art. 34, 0 and Pdetermine the perpendicular bisecting AB.

150. The two tangents from a point to a circle are equal.

Let PA and PB, Fig. 93, be fangents from P to the circle

whose center is 0. Draw O A, ГОР, ОВ. Then the triangles POB and POA are right triangles (Art. 148). In these triangles, PO is common and O A is equal to OB. Hence, the triangles are equal, and PA = PB.

FIG. 93

151. The line joining an external point to the center of a circle bisects the angle made by the two tangents drawn from the point to the circle. Thus, the angle OPA, Fig. 93, is equal to the angle OPB.

EXAMPLES FOR PRACTICE

1. Show that the line joining the intersection of two tangents to the center of the circle bisects the chord joining the points of tangency. 2. Show that the bisector of the angle between two tangents passes through the center of the circle.

FIG. 94

3. Show that in the same circle, or equal circles, equal chords are equally distant from the center.

SUGGESTION.-Draw OE and OF. Fig. 94, perpendicular to the equal chords A B and CD. Then what is true of the triangles A E O and DOF?

4. Show that the tangents to a circle at the extremities of a diameter are parallel.

5. Show that in any circle a chord parallel to a tangent is bisected by the diameter drawn

to the point of contact.

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GEOMETRY

MEASUREMENT OF ANGLES

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152. The ratio of one quantity to another of the same kind is the number of times that the first contains the second. When both quantities are represented by numbers, their ratio is the same as the quotient obtained by dividing one of the numbers by the other.

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TEXTBOOK COMPANY.

153. In the same circle, or equal circles, two central angles have the same ratio as their intercepted arcs; that is, in Fig. 95, angle A OB: angle COD arc AB a

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Suppose the arc A B to be three-fifths of the arc CD. Divide into three equal parts, and CD into five equal parts, as shown, join the points of division with the center.

Since A B C D = 3:5, or

AB 3
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CD 5'

that one-third of A B is one-fifth of CD; that is, arc A E = arc D F, and, therefore, angle AOE = angle DOF. We have, therefore, angle A OB = 3 × angle A O E, angle COD = 5 X angle DOF= 5 X angle A O E; whence, angle AOB 3X angle AOE_3 arc AB angle COD 5 X angle AOE 5 arc CD'

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E

B

FIG. 95

154. Since the angle at the center and its intercepted arc increase and decrease in the same ratio, it is said that an angle at the center is measured by its intercepted arc.

155. The whole circumference of a circle is divided into 360 equal parts, called degrees. A degree is divided into 60 equal parts, called minutes; and a minute is divided

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D

B

into 60 equal parts, called seconds. Degrees, minutes, and seconds of arc are used as units for measuring circular arcs. Since the circumference of every circle contains 360 degrees, the length of a degree differs in different circles. Thus, if AOB, Fig. 96, is an angle of 1°, AB is an arc of 1° in the larger circle and CD is also and arc of 1o in the smaller concentric circle. A degree of the earth's equator is a little more than 69 miles

FIG. 96

long; and a degree of the circumference of a circle whose diameter is 360 inches is 3.1416 inches long.

Degrees, minutes, and seconds are indicated by °, ',". Thus, 25° 3' 10" means 25 degrees, 3 minutes, and 10 seconds. Since a right angle intercepts one quarter of a circum-` ference, the number of degrees measuring it is 360 ÷ 4 90°. The number of degrees measuring an angle equal to one-half of a right angle is 90° ÷ 2 = 45°.

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Usually, the magnitude of an angle is expressed by stating the number of degrees that it subtends. Thus, a right angle is referred to as an angle of 90°; one-third of a right angle, as an angle of 30°, etc.

156. An inscribed angle is measured by one-half the intercepted arc. Thus, in Fig. 97, the angle ABC is measured by one-half the arc ADC.

Draw the diameter BOD and the radii OC and OA. The angle COD, the exterior angle of the triangle OBC, is equal to the angle OBC plus the angle OCB. But the angle OCB is equal to the angle OBC, as they are opposite the equal sides of an isosceles triangle. Hence, the angle COD, which is measured by the arc CD, is equal to 2X OBC. Therefore, OBC is measured by one-half the arc CD. Similarly, the angle OBA is measured by one-half the arc AD. Therefore, the angle ABC is measured by one-half the arc AD plus one-half the arc DC; that is, by one-half the arc A C.

FIG. 97

157. In the same circle, or equal circles, equal arcs are intercepted by equal inscribed angles.

158. All angles inscribed in the 4 same segment are equal.

159. Any angle inscribed in a semicircle is a right angle.

FIG. 98

B

The angle A CB, Fig. 98, is measured by one-half the arc A D B, which is a semicircumference. As a semi-circumference contains 180°, the angle A CB is measured by one-half of 180°, or 90°, and is, therefore, a right angle.

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