118 If one angle of a parallelogram is 82°, how many degrees in each of the other angles? 119 Two adjacent sides of a parallelogram are 14 feet and 23 feet respectively. Find the perimeter. 120 If one side of a rhombus is 6 inches in length, what is the length of the perimeter? 121 If two adjacent angles of a parallelogram are in the ratio 3:5, how many degrees in each angle? 122 If one angle of a parallelogram is a right angle, the figure is a right parallelogram. 123 If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. [▲ AOD = ▲ BOC (§ 162). .. AD = BC, and ≤1= ≤2 (§ 166). .. AD is to BC (§ 125). .. ABCD is a □ (§ 205).] B 124 The diagonals of a square are perpendicular to each other, and bisect the angles. [In ▲ ABD, AB = AD, AO bisects BD (§ 206). .. AO is 1 to BD and bisects ≤ A (§ 174).] 125 The diagonals of a rhombus are perpendicular to each other, and bisect the angles. [Proof identical with that in Ex. 124.] 126 The diagonals of a rectangle are equal. [Rt. ▲ ABD =rt. A ABC (§ 163). .. BD = AC.] 127 The diagonals of a rhomboid are unequal. [ZA is obtuse, and B is acute. .. ZA><B. A ABD and ABC, BD > AC (§ 170).] D Whence in 128 The diagonals of a rectangle, not a square, are oblique to each other. [In AOD and AOB, OA is common, OD = OB 4 (§ 206), and AD > AB. :. ZAOD > ZAOB (§ 171). That is, AC and BD are oblique.] 129 The diagonals of a rhomboid are oblique to each other. [The proof is identical with that of Ex. 128.] 130 In the parallelogram ABCD, AF bisects A, BE E Prove that AF and BE bisect each other. .. ABFE is a □ (§ 205).] B F 力。 D = BF (Ex. 115). 208 If three parallel lines intercept equal segments on one transversal, they intercept equal segments on every transversal. HYPOTHESIS. AE and BF are transversals to the parallels AB, CD, EF, and for HD = DK (proved), ≤ t = ≤ t' (§ 112), H § 200 § 167 $166 Q. E. D. K 209 COROLLARY 1. If a line is parallel to the base of a triangle and bisects one side, it bisects the other side also. Let DE be to BC, and AD = DB. Draw HAK to BC. HK is also to DE (§ 118). Now HK, DE, and BC intersect equal segments on AB by hypothesis; .. they intersect equal segments on AC. That is, DE bisects AC. E..F 210 COROLLARY 2. If a line bisects two sides of a triangle, it is parallel to the third side, and is equal to half the third side. Let DE bisect AB and AC. Draw DF to BC, and EG to AB. DF passes through E (§ 209), and coincides with to BC. Again: EG bisects BC by by const. ... DE = BG = } BC. DE (§ 94). .. DE is § 209, and DBGE is a 211 COROLLARY 3. The median of a trapezoid is parallel to the bases, and is equal to half the sum of the bases. E D F -H C Let EF be the median. Draw EH || to B AD. EH is also || to BC (§ 118), and bisects DC (§ 208). .. EF coincides with EH and is | to AD and BC. EF bisects AC at G (§ 208). .. EG = AD (§ 210). .. EF = } (AD + BC). = BC, and GF = EXERCISES 131 Every straight line drawn through the middle point of a diagonal, and terminated by the sides, of a parallelogram is bisected at that point. [Draw FOE to AD, and apply § 208.] 132 In the figure of § 208, prove that EF CD = CD AB. - 133 Prove that the median of a trapezoid is equal to half the sum of the bases by Ex. 132. 134 If a line is parallel to the base of a trapezoid and bisects one leg, it bisects the other leg also. 135 If a line bisects two sides of a triangle, it bisects every line drawn from the third side to the opposite vertex. 136 The line bisecting two adjacent sides of a quadrilateral is parallel and equal to the line bisecting the other two sides. 137 The median of a trapezoid bisects both diagonals. 138 In a right triangle, the mid-point of the hypotenuse is equidistant from the three vertices. [Let b = c, and xy. Then p is | to z by § 210. .. p is to x by § 117... a = b = c (§ 186).] 139 In the parallelogram ABCD, E and F are the mid-points of AD and BC. Prove that AF and EC trisect the diagonal BD. [AFCE is a (§ 205). In the ▲ DAG, EH bisects DG (§ 209). Likewise FG bisects BH in the ▲ BCH.] b u H G B F POLYGONS IN GENERAL DEFINITIONS * 212 A triangle (trigon) is a polygon of three sides. A dodecagon is a polygon of twelve sides. A pentadecagon is a polygon of fifteen sides. 213 An equilateral polygon is a polygon which has all its sides equal. An equiangular polygon is a polygon which has all its angles equal. A convex polygon is a polygon in which each angle is A concave polygon is a polygon which Thus, ABCDEFG is a concave polygon and has one reflex (reëntrant) angle, D. G F B A polygon is considered convex unless otherwise stated. E 214 A polygon may be divided into triangles by drawing diagonals from any vertex to all the vertices not adjacent. The number of triangles into which any polygon may be thus divided is evidently equal to the number of sides of the polygon less two; and the number of diagonals thus drawn is three less than the number of sides of the polygon. * The student should now review §§ 131-133. PROPOSITION XXXV. THEOREM 215 The sum of the angles of a polygon is equal to two right angles taken as many times, less two, as the polygon has sides. § 214 of the polygon. The polygon may be divided into (n-2) A. = 2 rt. . = (n § 153 2) 2 rt. 4. 140 How many diagonals can be drawn from one vertex in a polygon of n sides? 141 What is the sum of the angles of a quadrilateral? of a hexagon? 142 How many degrees in each angle of an equiangular octagon ? 143 How many sides has a polygon, the sum of whose angles is 14 right angles? 8 straight angles ? 7 perigons? n right angles? 144 How many sides has an equiangular polygon, if the sum of five of its angles is 8 right angles? |