Plane and solid geometry . , 4. Z CBD ^\ CD. 5. .-. Z ABD — Z CBD oc 1 (arc BCD — CD), 6. .-. A ABC oc 1 ^. Reasons 1. § 54, 15. 2. §313. 3. §297. 4. §365. 5. § 362, h, 6. §309. Ex. 507. In the figure of § 378, if arc BC = 100^\ find the numberof degrees in angle ABC ; in angle CBD ; in angle CBE. Ex. 508. If tangents are drawn at the extremities of a chord whichsubtends an arc of 120°, what kind of triangle is formed ? Ex. 509. If a tangent is drawn to a circle at the extremity of achord, the mid-point of the subtended arc is equidistant from the chordand the tangent. Ex. 510. Solve Pr
Plane and solid geometry . , 4. Z CBD ^\ CD. 5. .-. Z ABD — Z CBD oc 1 (arc BCD — CD), 6. .-. A ABC oc 1 ^. Reasons 1. § 54, 15. 2. §313. 3. §297. 4. §365. 5. § 362, h, 6. §309. Ex. 507. In the figure of § 378, if arc BC = 100^\ find the numberof degrees in angle ABC ; in angle CBD ; in angle CBE. Ex. 508. If tangents are drawn at the extremities of a chord whichsubtends an arc of 120°, what kind of triangle is formed ? Ex. 509. If a tangent is drawn to a circle at the extremity of achord, the mid-point of the subtended arc is equidistant from the chordand the tangent. Ex. 510. Solve Prop. XXII by means of Prop. Observe that in the figure for Prop. XXIV any angle inscribedin segment BDC would be equal to angle ABC, 154 PLAXE GEOMETRY Propositiox XXV. Theorem 379. An angle formed by two secants intersecting out-side of a circumference, an angle formed by a secant anda tangent J and an angle formed by two tangents are eachmeasured by one half tlie difference of tlie intercepted Fig. 2. Fig. 3. I. An angle formed by two secants (Fig. 1).Given two secants BA and BC, forming prove that Z1 ^ |- (ic — BE). Argument 1. Draw CD. 2. Z1 + Z2 = Z3. 3. .Z1 = Z3-Z2. 4. Z3 9^ i AC, and ^\DE. 5. .-. Z3-Z2 oc i.(jxr-^). 6. .•. Z 1 OC -1- (ic - BE). II. An angle formed by a secant and a tangent (Fig. 2). III. An angle formed by two tangents (Fig. 3).The proofs of II and III are left to the student. 380. Note. In the preceding theorems the vertex of the anglemay be : (1) within the circle; (2) on the circumference; (3) outsidethe circle. Reasons 1. § 54, 15. 2. § 215. 3. § 54, 3. 4. § 365. 5. §362,6. 6. § 309. Ex. 511. Tell how to measure an angle having its vertex in each ofthe three possible positions with regard to the circumference. BOOK II 155 Proposition XXVI. Theorem 381. Parallel lines intercept equal arcs on a circum-ference. A ^,7-^ B
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