Q.E.D. exercise. Pts.. . exercises. Rt. exterior. Scho. figure. Sq. figures. St. postulate. point. right. square. straight. hypothesis. Sup.. supplement. quod erat demonstrandum, which was to be proved. Q.E.F.. quod erat faciendum, which was to be done. PLANE GEOMETRY BOOK I RECTILINEAR FIGURES PROPOSITION I. THEOREM 100 All straight angles are equal. HYPOTHESIS. The ABC and A'B'C' are any two straight 4. ABC = A'B'C'. PROOF Place the ABC on the ZA'B'C' so that the vertex on the vertex B', and the line BA along the line B'A'. Then BC will fall along B'C'. .. the ABC and A'B'C' coincide and are equal. B falls Post. 3 $ 94 § 89 Q. E. D. § 100 $ 71 Ax. 9 Whence all rt. are halves of equal st. 4, and .. equal. "The halves of equals are equal." 102 COROLLARY 2. Only one perpendicular can be drawn to a given line at a given point in the line. PROOF If two such perpendiculars could be drawn, we should have unequal right angles, which is impossible (§ 101). PROPOSITION II. THEOREM 103 When one straight line meets another, the sum of the two adjacent angles thus formed is equal to two right angles. B D HYPOTHESIS. The line AD meets the line BC at D. ADB + ≤ ADC = 2 rt. . 104 COROLLARY 1. The complements of equal angles are equal. 105 COROLLARY 2. The supplements of equal angles are equal. 106 COROLLARY 3. The sum of all the angles that can be formed at a point in a straight line, and on the same side of the line, is equal to two right angles. For their sum is equal to a straight angle, "A straight angle is equal to two right angles." V Ax. 13 $ 70 107 COROLLARY 4. The sum of all the angles that can be formed at a point is equal to four right angles. For their sum is equal to a round angle, "A round angle is equal to four right angles." 108 COROLLARY 5. If one of the angles formed by the intersection of two straight lines is a right angle, the others are right angles also. 109 DEFINITION. Supplementary adjacent angles are two adjacent angles whose sum is equal to a straight angle; as, ▲ a and b. M N 110 COROLLARY. The exterior sides of supplementary adjacent angles are in the same straight line. For a + b = a st. 2. .. MN is a st. line. § 109 "A straight angle is an angle whose sides extend in opposite directions from the vertex, forming one straight line." $ 69 111 DEFINITION. Vertical angles are the two angles on opposite sides of two intersecting lines; as, a and b; also, c and d. EXERCISES axb a 1 Find the complement of an angle of 27°. 2 Find the supplement of an angle of 27°. 3 Of what angle is 58° the complement? 4 Of what angle is 126° the supplement ? 5 How many degrees in an angle if it is twice its complement? if one-third its complement? 6 How many degrees in an angle if it is five times its supplement? if one-fourth its supplement ? 7 How many degrees in each of two complementary angles if the greater exceeds the less by 24° ? 8 How many degrees in each of two supplementary angles if the greater exceeds the less by 36° ? 9 Find an angle the sum of whose complement and supplement is 102°. 10 An angle is of a perigon. How many degrees in its complement? in its supplement? 11 The sum of three angles is a perigon. The first is three times the second, and the second is twice the third; how many degrees in each? 12 An angle is 40° less than its complement. What fraction is it of a right angle? of a straight angle? 13 Three angles, whose sum is a right angle, are in the ratio 2:3:5. How many degrees in each angle? 14 What angle is included between the hands of a watch at 3 o'clock ? at 5 o'clock ? at 8 o'clock ? 15 What angle does the minute hand of a watch describe in 12 min. ? in 28 min.? in 1 hr.? 16 What angle does the hour hand of a watch describe in 30 min. ? in 1 hr.? in 34 hr.? |