PROPOSITION III. THEOREM 72. Two triangles are equal if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other (s.a.s. = s. a. s.). Α Α' B Hyp. In AABC and A'B'C', To prove B' AB= A'B', BC = B'C', and ≤B = LB', Proof. Apply ▲ ABC to ▲ A'B'C' so that BC shall coincide Ex. 35. If two lines bisect each other, and their extremities are joined, four triangles are formed, of which those opposite each other are equal. 73. REMARK. The equality of lines and angles is usually proved by means of equal triangles. Ex. 36. If any point in the perpendicular bisector of a line is joi to the extremities of that line, the lines joining the point to the extre ties are equal. Ex. 37. The bisector of the vertical angle of an isosceles tria bisects the base. Ex. 38. If, upon the sides of an angle, equal distances be laid off fi the vertex, and the ends be joined to any point in the bisector of angle, these lines are equal. Ex. 39. If in the triangle ABC, Z A = Z B, and the points D and be taken in AC and BC so as to make ▲ ABD = 2 EAB, then BD = = Ex. 40. If two angles of a triangle are equal, the bisectors of th angles are equal. Ex. 41. If, from the ends of the base of an isosceles triangle, e distances be laid off on the same arms, and from their ends lines be dra to the opposite vertices, these lines are equal. 74. REMARK. · If the lines and angles whose equality is to proved are not parts of triangles, try to construct such triang Ex. 42. If in triangle ABC, AB = BC, then ▲ A = ▲ C. PROPOSITION IV. THEOREM 75. An exterior angle of a triangle is greater th either remote interior angle. Proof. Let E be the midpoint of BC. duce it its own length to F. Draw FC. Draw AE By joining the midpoint of AC to B, it follows in t anner that Ex. 43. If four points, A, B, C, and D, in a straight line be a point, E, without, then (1) ZABE><ACE, (2) ZABE>▲ CED, (3) ZABE (See diagram on next nare ) E a b c\d B 76. DEF. When two straight lines, AB and CD, are cut by a third straight line, EF, called a transversal, then the angles a, b, g, and h are called exterior angles, the angles c, d, e, and f are called interior angles, the angles a and e, b and f, c and g, and d and h are called corresponding angles, the angles c and f, and d and e are called alternate interior angles, and the angles b and g, a and h, are called alternate exterior angles. PARALLEL LINES 77. DEF. Parallel lines are lines which lie in the same plane and do not meet if produced indefinitely. AB and CD are parallel. 78. AXIOM 11. Two intersecting lines cannot be both parallel to a third straight line. 79. THEOREM. A -B Two straight lines which are parallel to a third straight line are parallel to each other. For if the two lines should meet, we would have two intersecting straight lines parallel to a third straight line, which contradicts Axiom 11. с PROPOSITION V. THEOREM 80. Two lines are parallel if a transversal lines makes the alternate interior angles equal Hyp. AC and DF are intersected by BE so that Proof. AC and DF either meet or are parallel. Then BEG is a triangle whose exterior ABE is a remote interior / BEG, which is impossible. Hence AC and DF are parallel. 81. SCHOLIUM. That the lines AC and DF cannot the side of A and D can be proved by the same method 82. REMARK. - In order to demonstrate that two 1 parallel, prove the equality of a pair of alternate interior Ex. 48. In the same diagram, if ▲ AHC=ZEIH, CD is paralle Ex. 49. Lines are parallel if a transversal makes the alternate |