PROPOSITION XVIII. THEOREM 358. An angle at the center of a circle is measured by its intercepted arc. E C B Given central AOB and AB intercepted by it; let COɛ be any unit (e.g. a degree), and let CE, intercepted by the unit ≤, be the unit arc. To prove the measure-number of AOB, referred to COE, equal to the measure-number of AB, referred to CE. I. If ZAOB and COE are commensurable. (a) Suppose that ZCOE is contained in ZAOB an integral number of times. ARGUMENT 1. Apply COE to AOB as a measure. Suppose that LCOE is contained in ZAOB r times. 2. Then is the measure-number of AOB r referred to LCOE as a unit. 3. Now the r equal central which compose ZAOB intercept r equal arcs on the circumference, each equal to CE. ..r is the measure-number of AB referred to CE as a unit. 4. 5. ..the measure-number of ZAOB, referred to LCOE as a unit, equals the measure-number of AB, referred to CE as a unit. Q.E.D. REASONS 1. § 335. 2. § 335. 3. § 293, I. 4. § 335. 5. § 54, 1. I. If ▲ 40B and ▲ COE are commensurable. (b) Suppose that COE is not contained in number of times. The proof is left as an student. AOB an integral exercise for the HINT. Some aliquot part of COE must be a measure of ZAOB. (Why?) Try } ZCOE, } ZCOE, etc. II. If AOB and COE are incommensurable. ply 1 to 4OB as many times as 2. AOF and COE are commensurable. ferred to COE as a unit, equals the 2. § 337. 3. § 358, I. ARGUMENT 4. Now take a smaller measure of COE. No matter how small a measure of COE is taken, when it is applied as a measure to AOB, the remainder, FOB, will be smaller than the taken as a measure. 5. Also FB will be smaller than the arc intercepted by the taken as a measure. 6. the difference between AOF and ZAOB may be made to become and remain less than any previously assigned, however small; and likewise the difference between AF and AB, less than the arc intercepted by the assigned Z. 7... AOF approaches REASONS 4. § 335. 5. § 294. 6. Args. 4 and 5. AOB as a limit, 7. § 349. 8. § 359. and AF approaches AB as a limit. approaches the measure-number of 9. But the measure-number of ZAOF is 10... the measure-number of ZAOB, referred to COE as a unit, equals the measure-number of AB, referred to CE as a unit. Q.E.D. 9. Arg. 3. 10. § 355. 359. If a magnitude is variable and approaches a limit, then, as the magnitude varies, the successive measure-numbers of the variable approach as their limit the measure-number of the limit of the magnitude. (This theorem will be found in the Appendix, § 597.) 360. Cor. In equal circles, or in the same circle, two angles at the center have the same ratio as their intercepted arcs. HINT. The measure-numbers of the angles are equal respectively to the measure-numbers of their intercepted arcs. Therefore the ratio of the angles is equal to the ratio of the arcs. Ex. 478. Construct a secant which shall cut off two thirds of a given circumference. Ex. 479. Is the ratio of two chords in the same circle equal to the ratio of the arcs which they subtend? Illustrate your answer, using a semicircumference and a quadrant. ∞ is 361. The symbol will be used for is measured by. the symbol of variation, and the macron (-) means long or length. Hence suggests varies as the length of 362. From § 336 it follows directly that: * : (a) In equal circles, or in the same circle, equal angles are measured by equal arcs; conversely, equal arcs measure equal angles. the sum (b) The measure of the difference of two angles is equal to difference of the measures of the angles. sum } (c) The measure of any multiple of an angle is equal to that same multiple of the measure of the angle. 363. Def. An angle is said to be inscribed in a circle if its vertex lies on the circumference and its sides are chords. 364. Def. An angle is said to be inscribed in a segment of a circle if its vertex lies on the arc of the segment and its sides pass through the extremities of that arc. *It is, of course, inaccurate to speak of measuring one magnitude by another magnitude of a different kind; but, in this case, it has become a convention so general that the student needs to become familiar with it. More accurately, in Prop. XVIII, the measurenumber of an angle at the center, referred to any unit angle, is the same as the measurenumber of its intercepted arc when the unit arc is the arc intercepted by the unit angle. 365. An inscribed angle is measured by one half its intercepted arc. I. Let one side of Z ABC, as AB, pass through the center of III. Let center o lie outside of Z ABC (Fig. 3). The proofs of II and III are left as exercises for the student. 4? of 25? What, then, is the measure of XBC? HINT. In Fig. 2, what is the measure of is the measure of Z ABC? In Fig. 3, what of 4 XBA? What, then, is the measure of ABC? |