EXERCISES 24 If AB and CD are parallel and 0 = 110°, how many degrees in s? 25 If AB and CD are parallel and p = 70°, how many degrees in v? 26 If AB and CD are parallel and Zo= 110°, how many degrees in <r? 27 If AB and CD are parallel, prove that n and v are supplementary. 28 If ≤ p = 75° and ≤r = 75°, are AB and CD parallel? 29 If Zo 120° and r = 60°, are AB and CD parallel? = 30 If AB and CD are parallel and m = 82°, how many degrees in each of the other seven angles? 31 X and Y are two parallel lines cut by two parallel transversals S and T. Prove a = r; Ll = Lg. 32 In Ex. 31 prove that c + 2 o = 2 rt. Æ. 33 If a straight line intersects two parallels: T a/b m/n 0 1 The bisectors of the alternate interior angles are parallel. [§ 125.] 2 The bisectors of the corresponding angles are parallel. [§ 126.] 3 The bisectors of the interior angles on the same side of the transversal are perpendicular to each other. [Ex. 21, § 124.] 34 Perpendiculars to two parallels are parallel. [§ 117, § 115.] 35 If AB is || to CD, and ZA ZC, prove = [Draw AE' || to CF and prove that AE || to CF. AE and AE' coincide.] 36 If two right angles have two sides parallel, the other two sides are parallel. A B 37 X and Y are parallel lines; prove that ≤C = Za +2b. [Through the vertex C draw a line parallel to X.] 38 In a line AB, points C and D are so taken A that AD = CB; prove AC = DB. [Ax. 4.] E F 39 Two lines are not parallel if they are respectively perpendicular to two non-parallel lines. POLYGONS DEFINITIONS 131 A polygon is a plane surface bounded by straight lines; as, ABCD. The sides of a polygon are the bounding lines, whose sum is the perimeter of the polygon. B D The angles of a polygon are the angles formed by its sides, and the vertices of the angles are the vertices of the polygon. Adjacent angles of a polygon are two angles having a common side; as, angles B and C, C and D, etc. 132 A diagonal of a polygon is the straight line joining two vertices not adjacent. 133 An exterior angle of a polygon is the angle included between one side and an adjacent side produced; as, angle EDF. TRIANGLES 134 A triangle is a polygon of three sides. AAA G E F 135 A right triangle is a triangle having a right angle; as, A and B. 136 An oblique triangle is a triangle having all its angles oblique; as, C, D, E, F, and G. 137 An acute triangle is a triangle having all its angles acute; as, C, D, and E. 138 An obtuse triangle is a triangle having an obtuse angle; as, F and G. 139 An equiangular triangle is a triangle having all of its angles equal; as, E. 140 An equilateral triangle is a triangle having all of its sides equal; as, E. 141 An isosceles triangle is a triangle having two of its sides equal; as, A, D, E, and G. 142 A scalene triangle is a triangle having no two of its sides equal; as, B, C, and F. 143 The hypotenuse of a right triangle is the side opposite the right angle; as, AC. 144 The legs of a right triangle are the sides including the right angle; as, AB and BC. B 145 The legs of an isosceles triangle are the two equal sides; as, DE and DF. 146 The base of a triangle is the side upon which the triangle is assumed to stand. Any side of a triangle may be assumed as the base. In the isosceles triangle the unequal side is always the base, unless otherwise stated. 147 The vertical angle of a triangle is the angle opposite the assumed base, and the other two angles of the triangle are called the base angles. 148 The vertex of a triangle is the vertex of the vertical angle. Thus, in the triangle GHK, if HK is taken as the base, then angle G is the vertical angle, the point G is the vertex of the H triangle, and angles H and K are the base angles. G K 149 A median of a triangle is a straight line drawn from the vertex of an angle to the middle point of the opposite side. Every triangle has three medians. 150 An altitude of a triangle is the perpendicular drawn from the vertex of an angle to the opposite side. Every triangle has three altitudes. When we speak of the altitude of a triangle, we mean the altitude upon the assumed base. 151 A bisector of an angle of a triangle is a straight line bisecting an angle and terminated by the opposite side. Every triangle has three angle-bisectors. PROPOSITION XI. THEOREM 153 The sum of the three angles of a triangle is equal to two right angles.* Draw MN through B to AC, and produce AB and CB, forming the angles a, b, and c. Then A = a,"corresponding of || lines," § 122 Adding, < A + Z B + Z C = Za + Z b + 2 c Ax. 1 156 COROLLARY 3. If two triangles have two angles of the one equal to two angles of the other, the third angles are equal. Each angle of an equilateral triangle is 157 COROLLARY 4. 60°. * This theorem, one of the most important in geometry, was discovered by Pythagoras. |