Plane and solid geometry . BOOK IX 423 Proposition V. Theorem 918. A spherical angle is measured by the arc of agreat circle having the vertex of the angle as a pole andintercepted by the sides of tlie angle, prolonged if nec-essary. P__ ^ p. Given spherical Z APB, with CD an arc of a great O whose poleis P and which is intercepted by sides PA and PB of Z APB, To prove that Z APB oc CD, Outline of Proof 1. Draw radii OP^ OC, and OD. 2. From P draw PR tangent to PA and PT tangent to PB, 3. Prove OC and OD each ± OP. 4. Prove OC II PR, OD II PT, and hence Z COZ) = Z. RPT. 5. But Z COZ> Gc cB;

---

Plane and solid geometry . BOOK IX 423 Proposition V. Theorem 918. A spherical angle is measured by the arc of agreat circle having the vertex of the angle as a pole andintercepted by the sides of tlie angle, prolonged if nec-essary. P__ ^ p. Given spherical Z APB, with CD an arc of a great O whose poleis P and which is intercepted by sides PA and PB of Z APB, To prove that Z APB oc CD, Outline of Proof 1. Draw radii OP^ OC, and OD. 2. From P draw PR tangent to PA and PT tangent to PB, 3. Prove OC and OD each ± OP. 4. Prove OC II PR, OD II PT, and hence Z COZ) = Z. RPT. 5. But Z COZ> Gc cB; .-. Z i?Pr, Z ^P5 9^ CZ). 919. Cor. I. e^ spherical angle is equal to the plane angle of the dihedral angle formed by the planes of thesides of the angle. 920. Cor. II. The sum of all the spherical angles ahouta point on tlie surface of a sphere equals four right angles. Ex. 1486. By comparison with the definitions of the correspondingterms in plane geometry, frame exact definitions of the following classesof spherical angles : acute, right, obtuse, adjacent, complementary, supple-mentary, vertical. Ex. 1487. Any two vertical spherical angles are equal. Ex. 1488. If one great circle passes through the pole of anothergrea

---

Size: 1493px × 1674px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No

Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912