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.). Hyp. In A ABC and A'B'C", AB= A'B', BC= B'C", and B= 2 B', Proof. Apply A ABC to A A'B'C" so that BC shall coincide with B'C". BA will take the direction of B'A', (LB= 2 B' by hyp.). (AB = A' B' by hyp.). .. A ABC and A'B'C" coincide. A ABC= = A A'B'C". Q.E.D. 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 joined to the extremities of that line, the lines joining the point to the extremities are equal. Ex. 37. The bisector of the vertical angle of an isosceles triangle bisects the base. Ex. 38. If, upon the sides of an angle, equal distances be laid off from the vertex, and the ends be joined to any point in the bisector of the angle, these lines are equal. Ex. 39. If in the triangle ABC, ZA= ZB, and the points D and E be taken in AC and BC so as to make 2 ABD= EAB, then BD = AC. Ex. 40. If two angles of a triangle are equal, the bisectors of these angles are equal. Ex. 41. If, from the ends of the base of an isosceles triangle, equal distances be laid off on the same arms, and from their ends lines be drawn to the opposite vertices, these lines are equal. 74. REMARK. — If the lines and angles whose equality is to be proved are not parts of triangles, try to construct such triangles. Ex. 42. If in triangle ABC, AB = BC, then ZA= 2C. 75. An exterior angle of a triangle is greater than either remote interior angle. Proof. Let E be the midpoint of BC. Draw AE and produce it its own length to F. Draw FC. (hom. parts of equal A). But 2 BCD > L ECF. (Ax. 9.) Hence ZBCD > ZB. By joining the midpoint of AC to B, it follows in the same anner that LACG > LA. But LACG = BCD, (vertical 6). Hence Q.E.D. Ex. 43. If four points, A, B, C, and D, in a straight line be joined to a point, E, without, then (1) ZABE>LACE, (2) ZABE><CED, (3) LABE> LADE. (See diagram on next page.) Ex. 44. If in quadrilateral ABCD the diagonals AC and BD meet in E, find four angles smaller than angle BEA. Ex. 45. If from any point D in A ABC DA and DB are drawn, then <D><C. Hint. — Draw DC. all 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 A - cd 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. - B 78. AXIOM 11. Two intersecting lines cannot be both parallel cto a third straight line. 79. THEOREM. 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. PROPOSITION V. THEOREM 80. Two lines are parallel if a transversal to these lines makes the alternate interior angles equal. Hyp. AC and DF are intersected by BE so that ZABE = 2 BEF. AC || DF. Then BEG is a triangle whose exterior LABE is equal to a remote interior Z BEG, which is impossible. Hence AC and DF are parallel. Q.E.D. 81. SCHOLIUM. That the lines AC and DF cannot meet on the side of A and D can be proved by the same method. 82. REMARK. — In order to demonstrate that two lines are parallel, prove the equality of a pair of alternate interior angles. Ex. 46. In the annexed diagram, if ZIHC = 120°, and ZFIH = 120°, HV is CD parallel to EF ? Ex. 47. In the same diagram, Eif ZDHI = 60°, and ZFIH = 120°, CD is parallel to EF. Ex. 48. In the same diagram, if Z AHC=ZEIH, CD is parallel to EF. Ex. 49. Lines are parallel if a transversal makes the alternate exterior angles equal. |