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. B A E 100 The perpendiculars drawn to each of two parallels are parallel. B B 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. [41= 22 = 43.] 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 Ꭰ 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.] A A N M B M N C B Prove the 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. triangle OMN equilateral. Two cases. 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 nonparallel sides. The median of a trapezoid is the straight line joining the middle points of the legs. * The student should now review §§ 131-133. 198 The bases of a parallelogram are the side on which it stands and the opposite side, called its lower and upper bases. 199 The altitude of a parallelogram or trapezoid is the perpendicular distance between the bases. PROPOSITION XXIX. THEOREM 200 The opposite sides of a parallelogram are equal. 201 COROLLARY 1. A diagonal divides a parallelogram into two equal triangles. 202 COROLLARY 2. Parallels comprehended between parallels are equal. 203 COROLLARY 3. The opposite angles of a parallelogram are equal. PROPOSITION XXX. THEOREM 204 If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. HYPOTHESIS. ABCD is a quadrilateral, having AD = BC and AB = DC. "having three sides of each equal respectively." Iden. Hyp. Hyp. § 172 PROPOSITION XXXI. THEOREM 205 If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram. B HYPOTHESIS. ABCD is a quadrilateral, having AD equal and parallel to BC. CONCLUSION. ABCD is a parallelogram. 110 Two adjacent angles of a parallelogram are supplementary. 111 The bisectors of two adjacent angles of a paral lelogram are perpendicular to each other. [<α+ <b = art. 4. §§ 124, 153.] |