BOOK I LINES AND RECTILINEAR FIGURES PRELIMINARY THEOREMS 46. All right angles are equal. For all straight angles are equal (Ax. 15), and halves of equals are equal (Ax. 8). 47. At a given point in a given line only one perpendicular can be drawn to the line. For if two perpendiculars, AD and AC, could be drawn at A, we should have two unequal right angles BAD and BAC, which is impossible (46). E C B 48. Complements of the same angle or of equal angles are equal. (Ax. 3.) 49. Supplements of the same angle or of equal angles are equal. 50. If two adjacent angles have their exterior sides in a straight (Ax. 3.) line, they are supplementary. B (Ax. 10.) 51. If two adjacent angles are supplementary, their exterior sides are in the same straight line. Their exterior sides form a straight angle (36), and hence lie in a straight line (25). 52. The sum of all the angles about a point in a plane is equal to two straight angles. A मैं E Ex. 74. In the annexed diagram, if ABC is a straight line, and 3 is the supplement of 2, why is 21 = 23? Ex. 75. If in the annexed diagram, AB 1 BC, A'B' 1 B'C', and ≤3 = 24, prove ≤1 = 22. 3 B Ex. 76. If AB 1 CD and 21 = 22, prove 23=24. Ex. 77. If ABC is a right angle, and A is the complement of 1, prove ▲ A = 22. Ex. 78. If ABC is a st. 2, 21 = ≤ 2, and 3 and 4 are rt. 4, prove that 5 = 26. Ex. 79. If AB and CD are straight lines and 42 and 6 are right angles, prove that 23 = 25. Ex. 80. In the annexed diagram, if AB is a straight line, and ≤2 = 23, prove that 21 = 24. Ex. 81. If 21 is the supplement of ≤2, and ≤3 is the supplement of 22, why is 21 = 23? 3 STATEMENTS 21 is the sup. of 3. 2 is the sup. of 3. ..21=22. REASONS If two adj. have their ext. sides in a st. line, they are sup. For the same reason. Supplements of the same are equal. Q. E. D. 54. Every proof consists of a number of statements, each of which is supported by a definite reason. The only admissible reasons are: a previously proved proposition; an axiom; a definition; or the hypothesis. Ex. 82. In the diagram for the preceding proposition find ≤3 = 40°, (b) if ≤3 = m°. Ex. 83. Three straight lines, AD, BE, and CF, meet in 0, forming the six angles: a, b, c, d, e, and ƒ. (a) If Zb AOE. = 20°, and c = 60°, find (e) Ifb and ƒ are complementary, find ▲ d. (h) If ≤ AOC = 140°, and ≤ COE = 120°, find ▲ BOD. Ex. 84. In the same diagram prove that (a) 2b+2c = LAOE. FOB-Lc = La. (b) *Ex. 85. In the same diagram (a) If ZAOC = 150°, and ≤ COE = 130°, find ▲a. (b) If ▲ FOB = 140°, and ZAOC = 125°, find ≤ d. (c) If ZAOE + Z BOC = 140°, and ▲ c = 40°, find Ze. (d) If / FOB = FOD, prove that ▲f = e. (e) Prove that reflex ZAOE + reflex Z BOF+ reflex ZCOA = 720°. 55. DEF. A polygon is a portion of a plane bounded by straight lines. The lines are called the sides. The perimeter of a polygon is the sum of all its sides. The angles included by the adjacent sides are the angles of the polygon, and their vertices are the vertices of the polygon. An exterior angle is formed by a side and the prolongation of an adjacent one. A diagonal is a straight line joining two non-adjacent vertices. A E B * Exercises denoted by (*) are difficult and may be omitted in a first reading. C Thus ABCDE is a polygon of five sides, AB and BC are sides, an angle, point B is a vertex, and AC is a diagonal of the polygon. and 2 are exterior angles of the polygon. 56. DEF. A quadrilateral is a polygon of four sides. TRIANGLES - PART I A is 21 57. A triangle is a polygon of three sides. 58. Triangles classified as to sides. triangle no two sides of which are equal. A scalene triangle is a An isosceles triangle is a triangle two sides of which are equal. An equilateral triangle is a triangle all sides of which are equal. ΔΔ SCALENE EQUILATERAL ISOSCELES 59. Triangles classified as to angles. A right triangle is a triangle one angle of which is a right angle. An obtuse triangle is a triangle one angle of which is an obtuse angle. An acute triangle is a triangle all angles of which are acute. An equiangular triangle is a triangle all angles of which are equal. ADAA RIGHT OBTUSE ACUTE EQUIANGULAR 60. DEF. The base of a triangle is the side on which the figure appears to stand. |