PROPOSITION III. THEOREM. 264. An inscribed angle is measured by half its intercepted arc. Given: An inscribed angle ABC intercepting the arc AC; Find 0, the center of the circle. Then 1o. If o lies in a side BC of Z ABC, join 40. 3o. If o lies without BA and BC, draw BOD, a diam. arc AC. Q.E.D. 265. DEFINITION. A segment of a circle is the figure contained by a chord and its arc. 266. COR. 1. All angles C, D, E, inscribed in the segment AEDCB of a circle are equal. For each is measured by half the arc AFB. E B F 267. COR. 2. An angle inscribed in a semicircle is a right angle. For it is measured by a semicircumference; i.e., by a circumference, or a quadrant. 268. COR. 3. The arc intercepted by an inscribed angle is double the arc intercepted by an equal angle at the center. PROPOSITION IV. THEOREM. 269. The angle formed by a tangent and a chord meeting at the point of contact, is measured by half the intercepted arc. Given: An angle BAC formed by a tangent AB and a chord AC; To Prove: Angle BAC is measured by arc AC. But C is meas. by arc AD, .. ▲ ▲ is meas. by arc AD or 1⁄2 arc AC. Q.E.D. (195") (110") (264) PROPOSITION V. THEOREM. 270. The vertical angles formed by intersecting chords are each measured by half the sum of the intercepted arcs. B E D Given: Two chords AB, CD, intersecting in E; To Prove: Angles AEC, BED, are each measured by (arc AC+ arc BD). ▲ AEC or ▲ BED is meas. by (arc AC + arc BD). Q.E.D. EXERCISE 320. In the diagram for Prop. III., if B is an angle of 32°, how many degrees are there in arc AC? 321. In the diagram for Prop. IV., if CD is an arc of 1020, then BAC is an angle of how many degrees? 322. In the diagram for Prop. V., if AEC is an angle of 25° and AC an arc of 30°, how many degrees in arc BD ? 323. Any three points of a circumference being given, how can we find other points of it without knowing the center? 324. The opposite sides of an inscribed parallelogram divide in the same ratio the radii drawn perpendicular to them. 325. If two circles whose centers are O and O' have a common tangent AB, and OA, O'B, be joined, these lines will be parallel. 326. If two tangents, PA, PB, be drawn to a circle whose center is O, and AB, AO, be drawn, then will angle BAO = angle P. PROPOSITION VI. THEOREM. 271. The angle formed by two secants meeting without the circle is measured by half the difference of the intercepted arcs. B E D Given: Secants AB, AC, meeting in 4, and intercepting arcs BC, DE; To Prove: Angle A is measured by 1⁄2 (arc BC arc DE). EXERCISE 327. In the diagram for Prop. VI., if A is an angle of 170 and DE an arc of 36°, how many degrees in arc BC? 328. If a polygon of an even number of sides be inscribed in a circle, the sums of its alternate angles are equal. 329. Find a point equidistant from three given points. When is the problem impossible? 330. Under what conditions is it possible to find a point equidistant from four given points? 331. Prove the theorem given in Art. 155, by means of Prop. III., (171). 332. If an inscribed triangle has unequal angles, the greater angle intercepts the greater arc. 333. If two circles intersect, their line of centers produced will bisect each of the four arcs. PROPOSITION VII. THEOREM. 272. An angle formed by a tangent and a secant is measured by half the difference of the intercepted arcs. B Given Tangent AB and secant AC meeting in A and intercepting arcs BC, BD; To Prove: Angle A is measured by (arc BC-arc BD). 273. COR. The angle formed by two tangents is measured by half the difference of the intercepted arcs. EXERCISES. QUESTIONS. 334. If an angle A is measured by two thirds of a quadrant, and an angle B = 50°, what is the ratio of A to Z B? 335. By what fraction of a quadrant is the vertical angle of an isosceles triangle measured, (1) if it is twice as great as a base angle? (2) if it is as great? (3) if it is th as great? n |