22. If, in the annexed diagram, AO is perpendicular to CO, and BO is perpendicular to DO, find AOD, (a) if COB = 40°, (b) if COB = m°. 23. What relation exists between AOD and BOC in the preceding exercise ? 24. What angle is formed by the bisectors of a pair of adjacent supplementary angles ? 25. Three lines meet in a point, O, forming six angles, 1, 2, 3, 4, 5, and 6. Find angle 3, if angle 1 80° and angle 5 = 60°. GENERAL TERMS 42. A theorem is a geometrical truth requiring demonstration. 43. An axiom is a geometrical truth assumed as self-evident. 44. A problem is a question to be solved. 45. A proposition is a general term for a theorem or a problem. 46. A corollary is a theorem easily derived from another theorem. 47. A scholium is a remark. 48. A postulate is a construction so simple that its possibility is admitted without further demonstration. 49. The hypothesis is that which is assumed in the statement of a theorem. 50. The conclusion is that which follows from the hypothesis. 51. A proposition is the converse of another, when the hypothesis and the conclusion of the one are respectively the conclusion and the hypothesis of the other. AXIOMS 1. Things that are equal to the same or equal things are equal to each other. 2. If equals be added to equals, the sums are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. If equals be added to unequals, the sums are unequal. 5. If equals be subtracted from unequals, or unequals from equals, the remainders are unequal. 6. Doubles of equals are equal. 7. Halves of equals are equal. 8. The whole equals the sum of its parts. 9. The whole is greater than any of its parts. 10. A straight line is the shortest distance between two points. (For Axiom 11, see page 17 on parallel lines.) POSTULATES 1. A straight line can be drawn between any two points. 2. A straight line can be produced indefinitely. Hyp. Angles ABC and DEF are straight angles. To prove LABC: = DEF. Proof. Apply Z DEF to ZABC so that the vertex E coincides with the vertex B, and ED coincides with BA. Then EF will fall on BC (straight lines coinciding in part coincide throughout). Hence DEF = L ABC. 53. All right angles are equal. Q.E.D. (Ax. 7.) 54. At a given point in a given line there can be but one perpendicular to the line. (53) 55. The complements of the same or of equal angles are equal. (Ax. 3.) A 56. The supplements of the same or of equal angles are equal. (Ax. 3.) 57. If two adjacent angles have their exterior sides in a straight line, these angles are supplementary. (Ax. 8.) 58. The sum of all the angles formed at a point in a plane is equal to two straight angles. C B K H * A figure formed by straight lines only is called a rectilinear figure. PROPOSITION I. THEOREM 59. Vertical angles are equal. B Hyp. Angles 1 and 2 are vertical angles. (two adjacent angles whose exterior sides are in a straight line are supplementary). .. 21=22, (supplements of equal & are equal). Q.E.D. Ex. 26. If, in the diagram for above proposition, ZAOC is 80°, find the other angles. Ex. 27. If, in the same figure, ZAOB be bisected, and the bisector be produced through O, prove that COD is also bisected. Ex. 28. If three lines, AB, CD, and EF, meet in a point, O, prove LAOE-LFOD = LAOC. Ex. 29. In the same diagram, prove: E B F 60. DEF. A polygon is a portion of a plane bounded by three or more straight lines, which are termed sides, and whose sum is the perimeter of the polygon. 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 the vertices of two non-adjacent angles. 61. A polygon of three sides is called a triangle; one of four sides, a quadrilateral. TRIANGLES 62. A triangle having three unequal sides is a scalene triangle. An isosceles triangle has two of its sides equal. An equilateral triangle has its three sides equal Scalene Equilateral Isosceles 63. A triangle is called acute if all its angles are acute; right, if one of its angles is a right angle; obtuse, if one of its angles is obtuse. A triangle is called equiangular if all its angles are equal. Right Obtuse Acute Equiangular 64. The base of a triangle is the side on which the figure is supposed to stand. The base of an isosceles triangle is that side which is equal to no other; the two equal sides are called the arms. 65. The vertical angle of a triangle is the angle opposite the base. |