Ex. 60. If in the diagram for Prop. VIII the transversal be produced through B and E, the figure will contain three angles equal to ZABE. Find these angles. Ex. 61. If the opposite sides of a quadrilateral are parallel, they must be equal. PROPOSITION IX. THEOREM 89. If two parallel lines are cut by a transversal, the corresponding angles are equal. [Converse of Prop. VI.] E B Hyp. Two parallel lines CD and EF are intersected by AB in H and I, respectively. To prove LAHD=L HIF. ZAHD=/ CHI, (vertical). ▲ CHI=/ HIF, (alt. int. of || lines). LAHD / HIF. (Ax. 1.) Q.E.D. Ex. 62. In the diagram for Prop. IX, if ZAHD = 50°, how many degrees are in & EIB, CHI, AIF, and EIA? Ex. 63. In the same diagram, prove Ex. 66. If three points A, B, and C be joined, and BC be produced to D, Ex. 67. In the same diagram, if CE bisects ACD, then LA=LB. Ex. 68. The bisectors of supplementary adjacent angles are perpendicular to each other. PROPOSITION X. THEOREM 90. If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary. Hyp. Two parallel lines CD and EF are intersected by AB in H and I, respectively. (two adj. whose ext. sides are in a st. line are supplementary). .. 2 DHI+Z HIF = 2 rt. . Q.E.D. Ex. 69. In the diagram for Prop. IX, if CD is parallel to EF, prove that LAHD +2 HIE = 2 rt. . Ex. 70. If two parallel lines are cut by a transversal, the exterior angles on the same side of the transversal are supplementary. Ex. 71. The bisectors of a pair of corresponding angles, formed by parallel lines, are parallel. Ex. 72. If upon each of two parallel lines a perpendicular be erected at any point, these perpendiculars either coincide or are parallel. Ex. 73. In the annexed diagram, if AB is parallel to ED, and ZA ZD, prove that AC is parallel to DF. A E Ex. 74. If in the annexed diagram 4 A, B, and C are right angles, LD is also a right angle. B A Ex. 77. The bisectors of two interior angles on the same side of a transversal to two parallel lines are perpendicular to each other. Ex. 78. State and prove the converse of Ex. 71. PROPOSITION XI. THEOREM 91. Angles whose corresponding sides are parallel are either equal or supplementary. Hyp. To prove D A 4 B ZB+ZA'B'D = 2 rt. 4. [The proof is left to the student.] C 92. SCHOLIUM. The angles are equal if both the corresponding pairs of sides lie in the same or in opposite directions from the vertex. Ex. 9. If two sides of a quadrilateral are equal and parallel, the other two sides must be parallel. Ex. 80. If two sides of a triangle are respectively parallel to two homologous sides of an equal triangle, the third side of the first must be parallel to the third side of the second. PROPOSITION XII. THEOREM 93. The sum of the angles of a triangle is equal to a straight angle. But ..ZA+ZB+20=ZA+/DAB+ZCAE. (Ax. 2.) Q.E.D. 94. COR. 1. In a triangle there can be only one obtuse or one right angle. 95. COR. 2. The acute angles of a right triangle are complementary. 96. COR. 3. If two triangles have two angles of the one respectively equal to two angles of the other, the third angles are equal. 97. COR. 4. Two triangles are equal, if a side, the opposite angle, and any other angle of the one are equal respectively to a side, the opposite angle, and any other angle of the other triangle (s. a. a. = s. a. a.). 98. COR. 5. From a point without a line there can be only one perpendicular to that line. 99. COR. 6. Each angle of an equiangular triangle is equal to sixty degrees. Ex. 81. If an angle of a triangle is (1) 40°, (2) m°, what is the sum of the other two angles? Ex. 82. If one angle of a triangle is equal to the sum of the other two, 1) how many degrees has that angle? (2) what is such a triangle called? Ex. 83. If two angles of a triangle are 60° and 40°, respectively, what is the angle formed by the bisectors of these angles? Ex. 84. Find each angle of a triangle if the second equals twice the first, and the third equals three times the first. Ex. 85. If two angles of a triangle are (1) 40° and 60°, (2) m° and no, find the other interior and the exterior angles of the triangle. Ex. 86. If an exterior angle of a triangle is three-fourths of a right angle, a remote interior angle one-half of a right angle, find the other interior angles. Ex. 87. If two lines are intersected by a transversal, and the bisectors of the interior angles on the same side of the transversal are perpendicular to each other, these lines are parallel. Ex. 88. The altitude upon the hypotenuse of a right triangle divides the right angle into two parts, which are respectively equal to the two acute angles of the right triangle. Ex. 89. Find the sum of the four angles of a quadrilateral. Ex. 90. If two angles of a triangle are equal, the bisector of the third し angle divides the figure into two equal triangles. L Ex. 91. Two right triangles are equal if the hypotenuse and an acute angle of the one are equal respectively to the hypotenuse and an acute angle of the other. Ex. 92. The altitudes upon the arms of an isosceles triangle are equal. |
