Plane and solid geometry . V. (See Ex. 172.) Ex. 172. Prove Prop. XVII by using the hint contained in § 158. Ex. 173. If two sides of a triangle are 14 and 9, between what limit-ing values must the third side be ? Ex. 174. If the opposite ends of any two non-intersecting line seg-ments are joined, the sum of the joining lines is greater than the sum ofthe other two lines. P Ex. 175. Given two points, Pand i?, and a line AB not passingthrough either. To find a point O,on AB, such that FO + OH shallbe as small as possible. This exercise illustrates the lawby which light is reflected from amirror
Plane and solid geometry . V. (See Ex. 172.) Ex. 172. Prove Prop. XVII by using the hint contained in § 158. Ex. 173. If two sides of a triangle are 14 and 9, between what limit-ing values must the third side be ? Ex. 174. If the opposite ends of any two non-intersecting line seg-ments are joined, the sum of the joining lines is greater than the sum ofthe other two lines. P Ex. 175. Given two points, Pand i?, and a line AB not passingthrough either. To find a point O,on AB, such that FO + OH shallbe as small as possible. This exercise illustrates the lawby which light is reflected from amirror. The light from the object,P, is reflected and appears to comefrom L, as far behind the mirror as P is in front of it. Ex. 176. If from any point within a triangle lines are drawnto the extremities of any side of the triangle, the sum of these lines isless than the sum of the other two sides of the triangle. Hint. Let ABC be the given triangle, D the point within. ProlongAD until it intersects BC at E. Apply Prop. 171. Note to Teacher. Up to this point all proofs given havebeen complete, including argument and reasons. In writtenwork it is frequently convenient, however, to have studentsgive the argument only. These two forms will be distinguishedby calling the former a complete demonstration and the latter,which is illustrated in Prop. XIX, argument only. It is often a sufficient test of a students understanding of atheorem to have him state merely the main points involved ina proof. This may be given in enumerated steps, as in , or in the form of a paragraph, as in Prop. XLIV. Thisform will be called outline of proof. BOOK I 61 ]Proposition XIX. Theorem 172. If two triangles httve two sides of one equal respec-tively to two sides of the other, hut the included angle ofthe first greater than the included angle of the second, thenthe third side of the first is greater than the third side ofthe second.
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