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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.
Ex. 74. If in the annexed diagram & A, B, and C are right angles, 2D is also a right angle.
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.
AB || A'B' and BC|| B'C'. To prove
ZB+ZA'B'D=2 rt. [. [The proof is left to the student.]
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. 79. 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.
Hyp. ABC is a triangle.
}(alt. int. 6 of I lines.) 2C= CAE)" ..ZA+ZB+ZC=LA+ZDAB+ZCAE. (Ax. 2.) But ZA+ZDAB+ZCAE= Z DAE= a st. Z.
..ZA+ZB+ZC= a st. Z. (Ax. 1.) 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. d. d. = 8. 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 equilateral triangle is equal to sixty degrees.
Ex. 81. If an angle of a triangle is (1) 40°, (2). mo, 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) mo 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 bisector 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 angle divides the figure into two equal triangles.
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. PROPOSITION XIII. THEOREM
100. An exterior angle of a triangle is equal to the sum of the two remote interior angles.
Hyp. ZACD is an exterior angle of A ABC.
Ex. 93. If two angles of a triangle are equal, the bisector of the remote exterior angle is parallel to the opposite side of the triangle. Ex. 94. If in the diagram for Prop. XIII DA is drawn, then
ZADB<LACB. Ex. 95. Prove Prop. XIII by means of (56).
Ex. 96. The sum of the three exterior angles of a triangle is equal to four right angles.
Ex. 97. If the sum of two exterior angles of a triangle is equal to three right angles, the triangle is a right one.
Ex. 98. If from a point without an acute angle, perpendiculars are drawn upon the sides of the angle, these perpendiculars include an angle equal to the original angle.
* Ex. 99. The bisectors of two exterior angles of a triangle include an angle equal to half the third exterior angle.
* Exercises denoted by (*) are difficult and may be omitted at a first reading.
PROPOSITION XIV. THEOREM
101. The base angles of an isosceles triangle are equal.
Hint. — Let AD be the bisector of ZBAC, and prove the equality of the two triangles.
102. COR. An equilateral triangle is equiangular.
Ex. 100. If the base of an isosceles triangle is trisected, the lines joining the points of division with the vertex are equal.
Ex. 101. The vertical angle of an isosceles triangle is : (1) 40°; (2) mo. Find the base angles.
Ex. 102. How many degrees are in each angle of an isosceles right triangle ?
Ex. 103. A base angle of an isosceles triangle is : (1) two-fifths of a right angle; (2) in right angle. Find the other angles.
Ex. 104. Each base angle of an isosceles triangle is one-half the remote exterior angle.
Ex. 105. The perpendiculars from the midpoint of the base upon the arms of an isosceles triangle are equal.
Ex. 106. The vertical angle of an isosceles triangle is : (1) 40°; (2) mo. Find the angle formed by the base, and an altitude upon an arm,