To prove Q.E.D. Hyp. Angles ABC and DEF are straight angles. Z ABC = Z DEF. Proof. Apply 2 DEF to Z ABC 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 Z DEF L ABC. 53. All right angles are equal. (Ax. 7.) 54. At a given point in a given line there can be but one perpendicular to the line. (53) D 55. The complements of the same or of equal angles are equal. (Ax. 3.) A -C 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 A angles are supplementary. (Ax. 8.) 58. The sum of all the angles formed at a point in a plane is K H equal to two straight angles. * A figure formed by straight lines only is called a rectilinear figure. B PROPOSITION I. THEOREM 59. Vertical angles are equal. B В 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, LAOC 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 2 COD is also bisected. Ex. 28. If three lines, AB, CD, and EF, meet in a point, 0, prove LAOE – ZFOD=LAOC. Ex. 29. In the same diagram, prove: E B LAOE + ZEOD - LAOD = 2 ZAOC. Ex. 30. In the same diagram, prove: LAOC + ZEOB + ZDOF = 2 rt. 6. Ex. 31. In the same diagram, prove : Α' LAOE + <COB + LEOD = 4 rt. 4. 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. 66. In a right triangle the side opposite the right angle is called the hypotenuse, and the other two sides, the arms. 67. The three perpendiculars from the vertices of a triangle to the opposite sides (produced if necessary) are called the altitudes of the triangle; the three bisectors of the angles are called the bisectors of the triangle; and the three lines from the vertices to the midpoints of the opposite sides are called the medians of the triangle. 68. Two triangles are equal when a side and two adjacent angles of the one are equal respectively to a side and two adjacent angles of the other. AB= A'B', ZA= L A', and ZB= LB'. Proof. Apply A ABC to A A'B'C" so that AB shall coincide with A'B'. BC will take the direction of B'C", (ZB = 2 B' by hyp.). (ZA= L A' by hyp.). C will fall upon A'C" or its prolongation. .. the point C falling upon both the lines B'C" and A'C", must fall upon the point common to both lines, namely, C". ... A ABC and A'B'C" coincide .. A ABC=A A'B'C'. Q.E.D. 69. NOTE. — This method of proof (superposition) is employed in fundamental propositions only. The student should place those parts upon each other whose equality is known, and, by successive steps, trace the position of the rest of the figure. 70. Note. — In order to facilitate the citing of propositions, the following abbreviation is suggested for the above proposition : a. s. a. = a.s.a. Similar abbreviations will be suggested for other propositions. 71. DEF. Polygons are mutually equiangular if their angles are respectively equal, and mutually equilateral if their sides are respectively equal. If two polygons are mutually equiangular, lines or angles similarly situated are called homologous lines or angles. Thus AB and A'B' (Prop. II) are homologous sides, C and C' homologous angles, the medians drawn from A and A' respectively homologous medians, etc. Ex. 32. If a perpendicular be erected at any point upon the bisector of an angle to meet the sides of the angle, two equal triangles are formed. Ex. 33. If through the midpoint of a straight line any line be drawn to meet the perpendiculars erected at the ends of the given line, two equal triangles are formed. Ex. 34. If a diagonal of a quadrilateral bisects those angles whose vertices it joins, the diagonal divides the figure into two equal triangles. |