Proposition 15. Theorem. 78. Conversely, if a straight line cut two other straight lines, (1) so as to make the exterior-interior angles equal, or (2) so as to make the two interior angles on the same side together equal to two right angles, then the two straight lines are parallel. (1) Hyp. Let the st. line EF cut the two st. lines AB, CD at G and A EGB=GHD. AB || to CD. E B H C H, so that To prove Proof. Since EGB = /GHD, (Hyp.) (2) Hyp. Let / BGH+<GHD = 2 rt. ≤ s. (75) Proposition 16. Theorem. 79. Straight lines which are parallel to the same straight line are parallel to one another. Hyp. Let the st. lines AB, CD A be each to the st. line PQ. To prove C AB || to CD. H P Proof. Draw any st. line EGH, cutting the lines in E, G, and H. E B NOTE. If PQ lies between AB and CD, the Proposition may be proved in a similar way, though in this case it scarcely needs proof; for it is inconceivable that two straight lines, which do not meet an intermediate straight line, should meet each other. EXERCISES. 1. Two straight lines AB, CD, bisect each other at O. Show that the straight lines joining AC and BD are parallel. 2. If a straight line meets two parallel straight lines, and the two interior angles on the same side are bisected; show that the bisectors meet at right angles. Proposition 17. Theorem. 80. Two angles having their sides parallel each to each, are either equal or supplementary. Hyp. Let AB be || to DH, and A D Then since the s AB and DE are cut by BC, .. ZABC = < DGC. And since the Is BC and MF are cut by DH, .'. ≤ DGC = / DEF. .. ZABCDEF. /DEF is the supplement of DEM. .. ABC is supplementary to ≤ DEM. F (177) (77) (Ax. 1) (25) Q.E.D. 81. COR. Since / DEF = < MEH, being verticals, (55) ... 2 ABC = 2 MEH, and is supplementary to HEF. 82. SCH. Two parallels are said to be in the same direction, or in opposite directions, according as they lie on the same side or on opposite sides of the straight line joining the vertices. Thus AB and ED, and also BC and EF, are in the same direction because they lie on the same side of BE. But BA and EH, and also BC and EM, are in opposite directions. Hence, when both pairs of parallel sides are either in the same direction, or in opposite directions, the angles are equal. But when two of the parallel sides are in the same direction, and the other two in opposite directions, the angles are supplementary. Proposition 18. Theorem. 83. Two angles having their sides perpendicular each to each, are either equal or supplementary. Proof. Draw AHL to AC, and AKL to AB. Then AH is || to ED, and AK is || to EF. Two lines to the same st. line are || (70). Because AH and AK are respectively || to ED and EF, and extend in the same direction, 84. SCH. If both angles are acute or both obtuse, they are equal; but if one is acute and the other obtuse, they are supplementary. TRIANGLES. 85. A triangle is a plane figure bounded by three straight lines. The three straight lines which bound a triangle are called its sides. Thus, AB, BC, CA, are the sides of the triangle ABC. The angles of the triangle are the angles formed by the sides with each other; as BAC, ABC, ACB. The ver tices of these angles are also called the D A vertices of the triangle. B 86. An exterior angle of a triangle is the angle formed between any side and the continuation of another side; as CAD. The angles BAC, ABC, BCA are called interior angles of the triangle. When we speak of the angles of a triangle we mean the three interior angles. 87. An equilateral triangle is one whose three sides are equal. 88. An isosceles triangle is one which has two equal sides. 89. A scalene triangle is one which has three unequal sides. 90. A right-angled triangle is one which has a right angle. The side opposite the right angle is called the hypotenuse, and the other two sides the legs. 91. An acute-angled triangle'is one which has three acute angles. It will be shown hereafter that every triangle must have at least two acute angles, |