PROPOSITION XXVI. THEOREM 187 Every point equidistant from the extremities of a line is in the perpendicular bisector of the line. HYPOTHESIS. CD is to AB, P is any point in CD, and PA = PB. "Two rt. A are equal if the hypotenuse and a leg of the one are equal respectively to the hypotenuse and a leg of the other." .. AD = DB, "being homologous sides of equal ▲." § 178 § 166 Q. E. D. 188 COROLLARY. Two points each equidistant from the extremities of a line determine the perpendicular bisector of the line. EXERCISES 94 If two isosceles triangles have the same base, the straight line joining their vertices bisects their common base at right angles. Two cases. 95 The triangles ABC and DBC are on opposite sides B of the same base BC, having AB Prove AD 1 to BC. = BD, AC DC. D 96 In a triangle GHK, ≤ G = 80°, ≤ H = 70°. What angle does the bisector of the K make with HG? PROPOSITION XXVII. THEOREM 189 Of two oblique lines drawn from the same point in a perpendicular and cutting off unequal distances from the foot, the more remote is the greater. "Every point in the perpendicular bisector of a line is equidistant from "If two of a ▲ are unequal, the sides opp. are unequal, and the greater 190 COROLLARY 1. side is opp. the greater ." § 180 Q. E. D. Only two equal straight lines can be drawn from a point to a straight line. 191 COROLLARY 2. Of two unequal oblique lines drawn from a point in a perpendicular, the greater cuts off the greater distance from the foot of the perpendicular. 192 Every point in the bisector of an angle is equidistant from the sides of the angle. HYPOTHESIS. BD bisects the angle ABC, P is any point in BD, PH and PK are perpendiculars respectively to AB and BC. "Two rt. A are equal if the hypotenuse and an acute equal respectively to the hypotenuse and an acute of the one are of the other." § 168 § 166 Q. E. D. 193 COROLLARY. Every point equidistant from the sides of an angle is in the bisector of the angle. EXERCISES 97 A line parallel to the base of an isosceles triangle makes equal angles with the legs. 98 In the triangle ABC, BA is produced to D making AD AC. Prove ≤ D < ▲ BCD. = 99 DE is parallel to BC, the base of an isosceles triangle. Prove 1 and 2 supplementary. 100 The perpendiculars drawn to each of two parallels are parallel. 101 AB and CD are equal and parallel lines. Prove that AD and BC bisect each other. 102 On the same base and on the same side of it, there can be but one equilateral triangle. 103 If from any point in the bisector of an angle, a line is drawn parallel to one side of the angle and intersecting the other side, the triangle thus formed is isosceles. [<1= 22 = 23.] 104 Show by a diagram that Cor. 2, Prop. XV, is not true if the word "homologous" be omitted. 105 Why, in Cor. 1, Prop. XV, is the word "homologous" not inserted before "acute angle" in the last line? 106 If the vertex A of the triangle ABC is joined to any two points in the base, as D and E, prove that ▲ ADB >< AEB > < ACB. B C E 107 Two isosceles triangles are equal if the legs and altitude of the one are equal respectively to the legs and altitude of the other. [Prove by superposition, making the equal altitudes coincide.] 108 In an equilateral triangle ABC, M and N are any two points in BC. OM is parallel to AB, and ON is parallel to AC. Prove the triangle OMN equilateral. Two cases. B M 109 DEF is an equilateral triangle. GE and GF bisect the angles E and F respectively. GH and GK are parallel respectively to DE and DF. Prove that EH = HK KF. QUADRILATERALS DEFINITIONS * 194 A quadrilateral is a polygon of four sides. A trapezium is a quadrilateral which has no two sides. parallel. A trapezoid is a quadrilateral which has two sides parallel, and two non-parallel. An isosceles trapezoid is a trapezoid whose non-parallel sides are equal. Trapezium Trapezoid Isosceles Trapezoid 195 A parallelogram is a quadrilateral whose opposite sides are parallel. A right parallelogram is one whose angles are right angles. An oblique parallelogram is one whose angles are oblique. PARALLELOGRAMS Rectangle Square Rhomboid Rhombus 196 A rectangle is a right parallelogram. 197 The bases of a trapezoid are the parallel sides, called its lower and upper bases. The legs of a trapezoid are the non parallel sides. The median of a trapezoid is the straight line joining the middle points of the legs. * The student should now review §§ 131-133. |