DEFINITIONS 217. An inscribed angle of a circle is one whose vertex lies in the circumference and whose sides are chords. 218. A segment of a circle is a portion of a circle bounded by an arc and its chord. 219. An angle is said to be inscribed in a segment if its vertex lies in the arc and its sides pass through the extremities of that arc. Ex. 345. How many degrees are there in a central angle which intercepts of the circumference? of a semicircumference? 220. An inscribed angle is measured by one-half the intercepted arc. Hyp. Angle ACB is inscribed in circle O. Το prove ACB is measured by arc AB. is a diameter of the circle. CASE II. The center lies within the angle. Then ACD is measured by arc AD, and BCD is measured by arc BD. or LACB is measured by arc AB. CASE III. The center lies without the angle. Q.E.D. 221. COR. I. Angles inscribed in the same segment, or in equal segments, are equal. 222. COR. 2. An angle inscribed in a semicircle is a right angle. Ex. 346. If in the diagram for Case I, C = 30°, how many degrees are in arc CB? = Ex. 347. If in the same diagram arc BC 3 arc AB, find ▲ C. Ex. 348. If in the diagram for Case II arc AC is, and arc BC is of the circumference, find Z ACB, ZACD. Ex. 349. If in the diagram for Case III, A is the midpoint of arc CD, and B is the midpoint of arc AD, how many degrees are there in ▲ ACB? Ex. 350. If a quadrilateral ABCD be inscribed in a circle, and the two diagonals be drawn, find all the angles in the figure, if arc AB = 80°, arc BC 110°, and arc CD = 90°. = Ex. 351. In the diagram of the preceding exercise, find four pairs of equal angles. Ex. 352. The opposite angles of an inscribed quadrilateral are supplementary. Ex. 353. If through one of the points of intersection of two equal circles a line be drawn to meet the circumferences, the extremities of that line are equidistant from the other point of intersection. PROPOSITION XIV. THEOREM 223. An angle formed by two chords intersecting within the circle is measured by onehalf the sum of the intercepted arcs. Hyp. Two chords AB and CD intersect in E. D Proof. Draw DB, and apply Prop. XIII. A C E B PROPOSITION XV. THEOREM 224. An angle formed by a tangent and a chord drawn from the point of contact is measured by half the intercepted arc. D B Hyp. AB is a tangent, and AC is a chord. To prove Proof. The CAB is measured by arc AC. Draw the diameter AD. rt. 2 DAB is measured by arc DCA. ZDAC is measured by arc DC. (Prop. XIII. = Ex. 354. If in diagram for Prop. XIV arc AD 60°, arc BA = 140°, and arc CB = 20°, find ▲ AEC and ZDAE. Ex. 355. In the same diagram, prove that ZADE equals EBC. Ex. 356. If two perpendicular chords intersect within the circle, the sum of a pair of opposite intercepted arcs is equal to a semicircumference. Ex. 357. Prove Prop. XV by demonstrating the equality of Z ADC and CAB. Ex. 358. If at the vertex of an inscribed square a tangent be drawn, what angle is formed by the tangent and adjacent side? Ex. 359. If, in Prop. XV, arc AC 2 arc CD, find ▲ CAB. = Ex. 360. A chord is parallel to a tangent drawn through the midpoint of the subtended arc. PROPOSITION XVI. THEOREM 225. An angle formed by two secants, or two tangents, or a tangent and a secant, intersecting without a circle, is measured by half the difference between the intercepted arcs. AAA B E B D B Hyp. AB and AD are secants drawn from an external point A. To prove ZA is measured by (arc BD – arc EC). HINT. AZ BCD - 2 CBA. H Ex. 361. If the angle formed by two tangents is 60°, how many degrees are in each of the intercepted arcs ? Ex. 362. If in the diagram for Case I, arc BD find arc EC. 100° and ZA = 20°, Ex. 363. If in the same diagram arc EC = 60° and arc_EB = arc BD arc CD, find ZA. Ex. 364. If an angle formed by a secant and a tangent is 20° and the greater of the intercepted arcs is 90°, how many degrees are in the other intercepted arc ? 226. SCHOLIUM. If we consider an arc which intersects the sides of an angle as positive when it turns its concave side toward the vertex, and negative when it turns the convex side toward the vertex, Props. XIII, XIV, XV, and XVI may be stated as follows: 227. THEOREM. If the sides of an angle (indefinitely produced) intersect or touch a circumference, the angle is measured by one-half the algebraic sum of the intercepted arcs. PROPOSITION XVII. THEOREM 228. Parallels intercept equal arcs on a circum ference. B D Hyp. AB and CD are two parallel chords. |