PROPOSITION XVII. THEOREM 171 If two sides of one triangle are equal respectively to two sides of another, but the third side of the first is greater than the third side of the second, the angle opposite the third side of the first is greater than the angle opposite the third side of the second. A HYPOTHESIS. BC > B'C'. In the ABC and A'B'C', AB = A'B', AC A'C', and CONCLUSION. <A><A'. PROOF The A is either equal to, less than, or greater than the ZA'. If ▲ A < ZA', BC < B'C'. Both these conclusions contradict the hypothesis that BC > B'C'. .. ZA > Z A'. § 162 § 170 Q. E. D. EXERCISES 52 If AD be drawn to the middle point of BC in the triangle ABC above, prove that ≤ ADC > < ADB. [§ 171.] 53 In the triangle ABC, AD is drawn to the middle point 172 Two triangles are equal if the three sides of the one are equal respectively to the three sides of the other. The A is either greater than, less than, or equal to the ZD. If <A> <D, BC > EF. If < A << D, BC < EF. Both these conclusions contradict the hypothesis § 170 "having two sides and the included angle of the one equal respectively to two sides and the included angle of the other." EXERCISES 54 ABC and DBC are two isosceles triangles on the same base BC. Prove that the triangles ADB and ADC are equal. § 162 Q. E. D. 55 Prove Ex. 54 by drawing the triangles on opposite sides of the base. 56 HK and MN bisect each other at P. Prove Δ ΗΡΜ = Δ ΚΡΝ. [§ 162.] 173 In an isosceles triangle, the angles opposite the equal sides are equal. B D HYPOTHESIS. ABC is an isosceles triangle, having AB = AC. B= C. "Two A are equal if the three sides of the one are equal respectively to the three sides of the other." .. < B = ZC, "being homologous of equal A.” § 172 Q. E. D. 174 COROLLARY 1. The straight line drawn from the vertex of an isosceles triangle to the middle point of the base bisects the vertical angle and is perpendicular to the base. 175 COROLLARY 2. An equilateral triangle is equiangular. EXERCISES 57 The exterior angles at the base of an isosceles triangle are equal. 58 The triangle formed by joining the middle points of the sides of an isosceles triangle is isosceles. [A 2 and 3 are equal by § 162; whence ▲ 1 is isosceles by § 166.] PROPOSITION XX. THEOREM 176 If two angles of a triangle are equal, the sides opposite are equal, and the triangle is isosceles. "Two rt. A are equal if a leg and an acute of the one are equal respectively to a leg and the homologous acute of the other." .. AB = AC, § 169 "being homologous sides of equal A.” § 166 Q. E. D. 177 COROLLARY. An equiangular triangle is equilateral. EXERCISES 59 HK is drawn parallel to BC the base of the isosceles triangle ABC. Prove that the triangle AHK is isosceles. [§§ 122, 176.] 60 If two exterior angles of a triangle are equal, the triangle is isosceles. 61 If the vertical angle of an isosceles triangle is 60°, the triangle is equilateral. 62 The sum of any two exterior angles of a triangle is greater than two right angles. 63 The exterior angle formed by producing the hypotenuse of a right triangle is obtuse. 64 If one angle of a triangle is equal to the sum of the other two, the triangle is a right triangle. 65 If a line drawn from the vertex of a triangle to the mid-point of the base is perpendicular to the base, the triangle is isosceles. [§ 163.] 66 The bisector of the vertical angle of an isosceles triangle bisects the base at right angles. [§ 162.] 67 If the bisector of an angle of a triangle is perpendicular to the opposite side, the triangle is isosceles. [§ 169.] 68 If two isosceles triangles have their vertical angles equal, they are mutually equiangular. 69 The exterior angle at the vertex of an isosceles triangle is twice an angle at the base. [§ 158.] 70 The bisector of the exterior vertical angle of an isosce les triangle is parallel to the base. [Ex. 69, § 126.] 71 Two equal straight lines drawn from the same point to the same straight line make equal angles with the line. [§ 173.] 72 The perpendiculars drawn from the mid-point of the base of an isosceles triangle to the legs cut off equal parts from the legs. [§ 168.] 73 If two points in the base of an isosceles triangle are equidistant from the extremities of the base, they are also equidistant from the vertex. [§§ 162, 166.] 74 The sum of the three exterior angles of a triangle (one at each vertex) is equal to four right angles. [§ 158.] 75 The triangle formed by the base of an isosceles triangle and the bisectors of the base angles is isosceles. [Ax. 9, § 176.] 76 In the triangle ABC, B = 2Z C. BD bisects B and meets AC at D. Prove ▲ DBC isosceles. |