CASE II. When the arcs are incommensurable (§ 181). B A In the circle ABC, let AOB and BOC be central angles, intercepting the incommensurable arcs AB and BC, respectively. Let the arc AB be divided into any number of equal parts, and let one of these parts be applied to the arc BC as a measure. Since AB and BC are incommensurable, a certain number of the parts will extend from B to C', leaving a remainder CC' less than one of the parts. Draw OC'. Then since the arcs AB and BC' are commensurable, we have ZAOB arc AB (§ 189, Case I.) Now let the number of subdivisions of the arc AB be indefinitely increased. Then the length of each part will be indefinitely diminished; and the remainder CC', being always less than one of the parts, will approach the limit 0. Now, ZAOB ZBOC' always equal, and approach the limits arc AB are two variables which are respectively. By the Theorem of Limits, these limits are equal. (§ 188.) 190. SCH. The usual unit of measure for arcs is the degree, which is the ninetieth part of a quadrant (§ 146). The degree of arc is divided into sixty equal parts, called minutes, and the. minute into sixty equal parts, called seconds. If the sum of two arcs is a quadrant, or 90°, one is called the complement of the other; if their sum is a semi-circumference, or 180°, one is called the supplement of the other. 191. COR. I. By § 154, equal central angles, in the same circle, intercept equal arcs on the circumference. Hence, if the angular magnitude about the centre of a circle be divided into four equal parts, each part will intercept one-fourth of the circumference; that is, A right central angle intercepts a quadrant on the circumference. 192. COR. II. By § 189, a central angle of n degrees bears the same ratio to a right central angle as its intercepted arc bears to a quadrant. 90 But a central angle of n degrees is of a right central angle. 90 Hence, its intercepted arc is of a quadrant, or an arc of n degrees. The above principle is usually expressed as follows: This means simply that the number of degrees in the angle is equal to the number of degrees in its intercepted arc. PROPOSITION XXI. THEOREM. 193. An inscribed angle is measured by one-half its intercepted arc. CASE I. When one side of the angle is a diameter. C Let AC be a diameter, and AB a chord, of the circle ABC. To prove that BAC is measured by arc BC. (§ 192.) Therefore, But, BOC is measured by arc BC. Whence, is measured by arc BC. CASE II. When the centre is within the angle. Let AD be a diameter of the circle ABC. To prove that the inscribed angle BAC is measured by arc BC. Therefore, the sum of the angles BAD and CAD is measured by one-half the sum of the arcs BD and CD. Hence, ▲ BAC is measured by arc BC. CASE III. When the centre is without the angle. Let AD be a diameter of the circle ABC. To prove that the inscribed angle BAC is measured by arc BC. Now, and BAD is measured by arc BD, ($ 193, Case I.) Therefore, the difference of the angles BAD and CAD is measured by one-half the difference of the arcs BD and CD. Hence, BAC is measured by arc BC. 194. SCH. As explained in § 192, this proposition means simply that the number of degrees in an inscribed angle is one-half the number of degrees in its intercepted arc. 195. COR. I. All angles inscribed in the same segment are equal. Thus, if the angles A, B, and C are inscribed in the segment ADE, then each angle is measured by arc DE. (§ 193.) Whence, AZ B = Z C. = B E 196. COR. II. An angle inscribed in a semicircle is a right angle. Let BC be a diameter of the circle ABD, and BAC an angle inscribed in the semicircle ABC. To prove BAC a right angle. LBAC is measured by arc BDC. But,arc BDC is a quadrant. (§ 193.) Whence, BAC is a right angle. PROPOSITION XXII. THEOREM. 197. The angle between a tangent and a chord is measured by one-half its intercepted arc. Let AE be tangent to the circle BCD at B, and let BC be a chord. ABC is measured by arc BC. To prove that Draw the diameter BD. Then BD is perpendicular to AE. (§ 170.) Now since a right angle is measured by one-half a semicircumference, ▲ ABD is measured by Also, Z CBD is measured by Hence, arc BCD. arc CD. (§ 193.) ▲ ABD – ≤ CBD is measured by That is, ABC is measured by In like manner, ▲ EBC is measured by arc BDC. (arc BCD arc CD). arc BC. |