ILLUSTRATIVE DEMONSTRATION THEOREM. If two straight lines intersect, the vertical angles are equal. Hypothesis. St. lines AB and CD intersect at O, forming vertical 1 and 2. Ex. 93. Prove in the same manner that ZAOD = ▲ COB. Ex. 94. If 3 = 130°, how many degrees are there in each of the other angles in the figure above? Ex. 95. Two lines intersect so that one of the angles is a right angle. How large is each of the other angles formed? 55. Besides the proofs of certain theorems, the methods of constructing certain figures are studied in geometry. A Problem is a construction to be made. 56. A Postulate is a construction assumed possible. The following postulates are necessary at the present time. POSTULATES 1. One straight line can be drawn through two points. (§ 8.) 2. A straight line can be extended indefinitely in each direction. (§ 8.) 3. A circle can be drawn with any point as center and any given segment as radius. 57. The word Proposition is commonly used to designate a theorem or a problem discussed in the text. SUPPLEMENTARY EXERCISES Ex. 96. Draw a line to represent the path of a baseball when the pitcher throws an out-curve." 66 He first places Ex. 97. A farmer is setting out trees for an orchard. the trees which are at the ends of a row. How may he then locate the other trees of that row so that they will be in a straight line without using a long line between the two end trees? What two points determine the straight line formed by the trees? Ex. 98. How do plasterers use in their work the characteristic property of a plane mentioned in § 1? Ex. 99. A man wishes a scale drawing of his suburban lot so that he may consult a landscape architect about the proper planting of it. He made the adjoining rough drawing of the lot, and then obtained the measurements indicated. Make a scale drawing of the lot, letting inch represent 25 feet. Ex. 100. Supplementary and Complementary Angles Ex. 101. What is the supplement of 56° 30' ? 150 275 200' 350' 100 Ex. 102. How many degrees are there in an angle if its complement contains 40° ? Ex. 103. How many degrees are there in an angle whose supplement contains 80° ? Ex. 104. Find the angle whose supplement is ten times its comple ment. Ex. 105. less by 25°. Ex. 106. Ex. 107. angle is 140°. Two angles are complementary. The greater exceeds the Find the angle which exceeds its supplement by 34°. The sum of the supplement and complement of a certain Ex. 108. Find the number of degrees in the angle the sum of whose supplement and complement is 196°. Ex. 109. The supplement of a certain angle exceeds three times its complement by 18°. Find the angle. Ex. 110. Prove that the supplement of any angle exceeds its complement by one right angle. Vertical Angles Ex. 111. Prove that the straight line which bisects an angle also bisects the vertical angle. OE bisects AOC. EOF is a st. line. Hyp. Con. OF bisects ▲ BOD. E A B Ex. 112. Prove that the bisectors of two supplementary adjacent angles are perpendicular. Suggestions.≤1 = } ≤ BCD; ≤2=}LDCA. E D F .. 21+22= ? A C B Ex. 113. If the bisectors of two adjacent angles are perpendicular, the angles are supplementary. (See figure of Ex. 112.) Ex. 114. If the bisectors of two adjacent angles make an angle of 45°, the angles are complementary. Ex. 115. Hyp. CO bisects ZAOB. Con. 23 = 24. Suggestions.-1. Compare 41 and 42. DEL CO. 2. Compare 43 and 1; 44 and 42. B E Ex. 116. Hyp. ZABC is a rt. Z. <3 is complementary to 1. Con. 23 = 22. D B Ex. 117. Two straight lines intersect so that one angle formed is 60°. How large is each of the other angles ? Ex. 120. What is the hypothesis of a theorem? What is the conclusion of a theorem ? Ex. 121. Lay off on a field some irregular piece of ground. Obtain such measurements as will enable you to make a scale drawing of the field. (This exercise is similar to Ex. 99.) Ex. 122. Another method for obtaining measurements for a scale drawing for a piece of ground and objects upon it is to locate the instrument for measuring angles at a point like O in the adjoining figure. Then find the distances of the other points from O and their directions from OD A or from OA. Barn Tree House Tree Tree Select an irregular piece of ground and obtain the measurements which will enable you to make a scale drawing of it and locate upon the drawing some of the trees or other objects on the lot. Supplementary Notes on Definitions NOTE 1.- Point and straight line are undefined. (See § 4 and § 5.) A definition describes a term by means of simpler terms. It is evident then that there must be some terms which cannot be defined, as there are no terms simpler than them by which to define them. No definition of point can be given. No satisfactory definition of straight line suitable for high school pupils can be given. Hence point and straight line are left undefined. NOTE 2.- Relating to the definition of an angle (§ 20).. 1 42 B Two rays OA and OB actually form two angles; namely, 1 and 2 of the adjoining figure. In 21, OA is the initial line (§ 20) and OB is the terminal line; in ≤2, OB is the initial line and OA is the terminal line. Usually, one angle is less than and the other is greater than a straight angle. Unless something is said to the contrary, ZAOB refers to the smaller angle formed by the rays OA and OB. |