Plane and solid geometry . outa 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 anothergreat circle, the circles are perpendicular to each other. 424 SOLID GEOMETRY Li:t^ES AKD PLAISTES TAXGEXT TO A SPHERE 921. Def. A straight line or a plane
Plane and solid geometry . outa 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 anothergreat circle, the circles are perpendicular to each other. 424 SOLID GEOMETRY Li:t^ES AKD PLAISTES TAXGEXT TO A SPHERE 921. Def. A straight line or a plane is tangent to a sphere if, however far extended, it meets the sphere in one and onlyone point. 922. Def. Two spheres are tangent to each other if theyhave one and only one point in common. They are tangentinternally if one sphere lies within the other, and externally ifneither sphere lies within the other. Proposition VL Theorem 923. A plane tangent to a sphere is perpendicular tothe radius drawn to the point of Given plane AB tangent to sphere 0 at T, and OT a radiusdrawn to the point of prove plane AB J. OT,The proof is left as an exercise for the student. 924. Question. What changes are necessary in the proof of § 313to make it the proof of § 923 ? 925. Cor. I. (Converse of Prop. VI). A plane perpen-dicular to a radius of a spJiere at its outer extremity istangent to the sphere^ Hint. See § 314. Ex. 1489. A straight line tanirent to a sphere is perpendicular to theradius drawn to the point of tangency. Ex. 1490. State and prove the converse of P2x. 1489. BOOK IX 425 Ex. 1491. Two lines tangent to a sphere at the same point determinea plane tangent to the sphere at that point. Ex. 1492. Given a point P on the surface of sphere 0. Explain howto construct: (a) a line tangent to sphere 0 at P; (?>) a plane tangentto sphere 0 at P. Ex. 1493. Given a point B outside of sphere Q. Explain how to con-struct : (a) a line through B tangent to sphere Q ; (6
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