SUMMARY 172. Truths established in Book I. 1. Two lines are equal, a. If they can be made to coincide. b. If they are sides of an equilateral triangle. j. If they are the sides of a square. i. If they are the non-parallel sides of an isosceles trapezoid. h. If they represent the distances of any point in the bisector of an angle from its sides. g. If they are drawn from any point in a perpendicular to a line and cut off equal distances on that line from the foot of the perpendicular. c. If they represent the distances from the extremities of a straight line to any point in the perpendicular erected at its middle point. d. If they are homologous sides of equal triangles. e. If they are sides of a triangle opposite equal angles. f. If they are sides of an equiangular triangle. § 36 § 91 § 103 § 108 § 118 § 119 § 132 § 134 § 139 k. If they are the sides of a rhombus. § 143 m. If they are opposite sides of a parallelogram. 7. If they are parallel and are intercepted between parallel lines. § 144 § 151 p. If one joins the middle points of the non-parallel sides of a trapezoid and the other is equal to half the sum of the parallel sides. o. If one is half a side of a triangle and the other is drawn parallel to it and bisecting one of the other sides. n. If they are parts intercepted on one transversal by parallel lines which intercept equal parts on another transversal. § 153 § 157 § 158 § 160 2. Two lines are parallel, a. If both are perpendicular to the same line. b. If when cut by a transversal the alternate interior angles are equal. 871 § 75 c. If when cut by a transversal the corresponding angles are equal. § 77 d. If when cut by a transversal the sum of the two interior angles on the same side of the transversal is equal to two right angles. e. If both are parallel to a third line. f. If they are the bases of a trapezoid. g. If they are opposite sides of a parallelogram. § 79 § 80 § 138 § 140 h. If one is a side of a triangle and the other joins the middle points of the other two sides. § 159 i. If one is either base of a trapezoid and the other joins the middle points of the non-parallel sides. 3. Two lines are perpendicular to each other, § 160 a. If they form equal adjacent angles with each other. b. If one is perpendicular to a line which is parallel to the other. § 26 $ 72 c. If any two or more points in one are each equidistant from the extremities of the other. §§ 106, 104 d. If one is the base of an isosceles triangle and the other is the bisector of the vertical angle. 4. Two lines form one and the same straight line, $ 120 a. If they are the sides of a straight angle. b. If they are the exterior sides of adjacent supplementary angles. $ 27 $ 58 5. Two lines are unequal, c. If they are sides of a triangle and lie opposite unequal angles. b. If they represent the distances from the extremities of a straight line to any point without the perpendicular erected at its middle point. a. If one is a perpendicular from a point to a straight line and the other is any other line from that point to the straight line. § 61 d. If they are the third sides of two triangles whose other sides are equal, each to each, and include unequal angles. § 103 § 127 § 129 e. If they are drawn from any point in a perpendicular to a line and cut off unequal distances on that line from the foot of the perpendicular. § 132 f. If they are distances cut off on a line from the foot of a perpendicular to it by unequal lines from any point in the perpendicular. $ 133 g. If one is any side of a triangle and the other is equal to the sum of the other two sides. §§ 124, 125 h. If one is equal to the sum of two lines from a point within a triangle to the extremities of one side, and the other is equal to the sum of the other two sides. § 131 6. A line is bisected, C. If they are the perpendicular bisectors of the sides of a triangle. § 170 d. If they are perpendiculars from the vertices of a triangle to the opposite sides. a. If it is the base of an isosceles triangle, by the bisector of the vertical angle. § 120 b. If it is the base of an isosceles triangle, by a perpendicular from the vertex. § 122 c. If it is either diagonal of a parallelogram, by the other diagonal. § 154 d. If it is the side of a triangle, by a straight line drawn parallel to the base and bisecting the other side. 7. Lines pass through the same point, a. If they are the medians of a triangle. b. If they are the bisectors of the three angles of a triangle. § 158 § 168 § 169 § 171 8. A perpendicular, and only one, can be drawn to a straight line, a. At a point in the line. b. From a point without the line. § 51 $ 60 g. If they are alternate interior angles formed by a transversal and parallel lines. $ 73 h. If they are corresponding angles formed by a transversal and parallel lines. § 76 i. If their sides are parallel and both pairs extend in the same or in opposite directions from their vertices. k. If they are angles of an equiangular triangle. j. If their sides are perpendicular to each other and both angles are acute or both are obtuse. $ 81 n. If they are homologous angles of equal triangles. m. If they are formed by the perpendicular bisector of a straight line and lines from any point in it to the extremities of the straight line. 7. If they are formed adjacent to a straight line by lines joining the extremities of that line with any point in its perpendicular bisector. § 83 § 98 $ 105 § 105 equal, each to each. o. If they are the third angles of two triangles whose other angles are § 108 q. If they are angles of an equilateral triangle. p. If they are opposite the equal sides of an isosceles triangle. § 113 § 116 r. If they are the opposite angles of a parallelogram. § 117 $ 149 10. Two angles are supplementary, a. If their sum is equal to two right angles. $ 32 a. If they are angles of a triangle and lie opposite unequal sides. b. If they are the angles opposite unequal sides of two triangles whose other two sides are equal, each to each. b. If their corresponding sides are parallel and one pair extends in the same direction and the other in opposite directions from their vertices. § 81 c. If their corresponding sides are perpendicular and one angle is acute and the other obtuse. 11. Two angles are unequal, § 83 $ 126 § 130 12. An angle is bisected, b. If it is the vertical angle of an isosceles triangle, by a line from the vertex perpendicular to the base. § 122 a. If it is the vertical angle of an isosceles triangle, by the perpendicular bisector of the base. § 121 angle. c. By a line every point of which is equidistant from the sides of the § 135 b. If they are the acute angles of a right triangle. 13. An angle is equal to the sum of two angles, a. If it is an exterior angle of a triangle, and the two angles are the opposite interior angles. 14. The sum of angles is equal to a right angle, a. If they are complements of each other. § 115 § 31 § 111 15. The sum of angles is equal to two right angles, a. If they are supplements of each other. b If they are adjacent angles formed by one straight line meeting another. $ 32 c. If they are all the consecutive angles which have a common vertex in a line and lie on the same side of the line. § 55 d. If they are the interior angles formed by a transversal and parallel lines and lie on the same side of the transversal. § 56 e. If they are the angles of a triangle. $78 § 110 16. The sum of angles is equal to four right angles, point. a. If they are all the consecutive angles that can be formed about a b. If they are the exterior angles of any convex polygon formed by producing the sides in succession. $ 57 § 167 17. The sum of angles is equal to (n − 2) 2 rt. 4, a. If they are the angles of any convex polygon. § 166 18. Two triangles are equal, a. If two sides and the included angle of one are equal to the corresponding parts of the other. b. If a side and two adjacent angles of one are equal to the corresponding parts of the other. § 100 c. If the three sides of one are equal to the three sides of the other. § 107 d. If they are right triangles, and a side and an acute angle of one are equal to the corresponding parts of the other. § 114 § 102 e. If they are right triangles, and the hypotenuse and a side of one are equal to the corresponding parts of the other. f. If they are formed by dividing a parallelogram by one of its diagonals. § 123 19. Two parallelograms are equal, $ 152 a. If they can be made to coincide. b. If two sides and the included angle of one are equal to the corresponding parts of the other. $ 36 $ 155 SUPPLEMENTARY EXERCISES Ex. 124. If through a point halfway between two parallel lines two transversals are drawn, they intercept equal parts on the parallel lines. Suggestions for Demonstration. 1. What are the data of the proposition ? 2. What lines and point in the figure in the margin represent the data of the proposition ? H A -B E 3. What parts of the figure are to be C proved equal? 4. How may two lines be proved equal? Ꮐ Summary, § 172, 1. 5. Since FH and JG, which are to be proved equal, are parts of triangles, what propositions might we expect to employ in the proof? 6. In what ways may two triangles be proved equal? Summary, § 172, 18. 7. What facts in the data suggest aid in determining the equality of angles? Ans. Parallel lines. 8. What homologous angles in the two triangles are equal? 9. What other homologous elements of the two triangles must also be equal before the triangles can be proved to be equal? 10. By careful examination of the given figure discover whether any two homologous sides can be proved equal. 11. Since the homologous sides cannot be proved equal from the given figure, if they can be proved equal at all, what expedient must be resorted to? Ans. Construction lines must be drawn which will enable us to prove a side of one of the triangles equal to an homologous side of the other. 12. What fact in the data has not yet been considered which might suggest aid in drawing the construction lines? 13. What kind of a line measures the distance between two parallel lines? If such a line be drawn through the given point, how is it divided at the given point? Then, what line may aid in the proof? 14. Drawing the figure as in the margin, with LK perpendicular to the parallel lines, 4and passing through the point E, which is halfway between the parallel lines, discover, how the triangles FEL and GEK compare ; also, how FE and GE compare. F Ꮮ H -B E J K G 15. Since the homologous angles of the original triangles have been discovered to be equal, and since the equality of two homologous sides, FE and GE, has also been shown, how do the original triangles compare? How do the sides FH and JG compare? Write out the demonstration, |