PROPOSITION XVII. THEOREM. 75. Two straight lines which are parallel to a third straight line are parallel to each other. Let A B and CD be parallel to E F. Since A B and E F are |, HK is also to A B, $ 67 (when two straight lines are cut by a third straight line, if the ext.-int. 4 be equal, the two lines are ||). Q. E. D. Let A B and CD be two parallel lines, and from any two points in AB, as E and H, let EF and HK be drawn perpendicular to A B. We are to prove EF=HK. Now EF and HK are 1 to C D, Draw MPL to A B. $ 67 On MP as an axis, fold over the portion of the figure on the right of MP until it comes into the plane of the figure on the left. and the point K will fall on E F, or E F produced. Also, PD will fall on PC, and the point K will fall on P C. Since the point K falls in both the lines E F and P C', it must fall at their point of intersection F. .. HK = EF, (their extremities being the same points). $ 18 Q. E. D. PROPOSITION XIX. THEOREM. 77. Two angles whose sides are parallel, two and two, and lie in the same direction, or opposite directions, from their vertices, are equal. 4.77 H C E F Fig. 1. H Fig. 2. E Let&B and E (Fig. 1) have their sides BA and E D, and BC and E F respectively, parallel and lying in the same direction from their vertices. Produce (if necessary) two sides which are not | until they Let&B' and E' (Fig. 2) have B' A' and E' D', and B' C' and E' F' respectively, parallel and lying in opposite directions from their vertices. = We are to prove the LB'LE'. Produce (if necessary) two sides which are not | until they intersect, as at H'. PROPOSITION XX. THEOREM. 78. If two angles have two sides parallel and lying in the same direction from their vertices, while the other two sides are parallel and lie in opposite directions, then the two angles are supplements of each other. Let A B C and D E F be two angles having BC and ED parallel and lying in the same direction from their vertices, while E F and B A are parallel and lie in opposite directions. We are to prove Z ABC and DEF supplements of each other. Produce (if necessary) two sides which are not until they intersect as at H. ZABC Z BHD, (being ext.-int. ). ZDEF BHE, (being alt.-int. ). $ 70 $ 68 But BHD and ▲ BH E are supplements of each other, § 34 (being sup.-adj. ). ZABC and DEF, the equals of BHD and LBH E, are supplements of each other. Q. E. D. ON TRIANGLES. 79. DEF. A Triangle is a plane figure bounded by three straight lines. A triangle has six parts, three sides and three angles. 80. When the six parts of one triangle are equal to the six parts of another triangle, each to each, the triangles are said to be equal in all respects. 81. DEF. In two equal triangles, the equal angles are called Homologous angles, and the equal sides are called Homologous sides. 82. In equal triangles the equal sides are opposite the equal angles. SCALENE. AA ISOSCELES. EQUILATERAL. 83. DEF. A Scalene triangle is one of which no two sides are equal. 84. DEF. An Isosceles triangle is one of which two sides are equal. 85. DEF. An Equilateral triangle is one of which the three sides are equal. 86. DEF. The Base of a triangle is the side on which the triangle is supposed to stand. In an isosceles triangle, the side which is not one of the equal sides is considered the base. |