66. In a right triangle the side opposite the right angle is called the hypotenuse, and the other two sides, the arms. 67. The three perpendiculars from the vertices of a triangle to the opposite sides (produced if necessary) are called the altitudes of the triangle, and the three lines from the vertices Altitude to the midpoints of the opposite sides are called the medians of the triangle. PROPOSITION II. THEOREM 68. Two triangles are equal when a side and two adjacent angles of the one are equal respectively to a side and two adjacent angles of the other. Proof. Apply AABC to AA'B'C" so that AB shall coincide with A'B'. BC will take the direction of B'C', (LB = LB' by hyp.). C will fall upon B'C' or its prolongation. AC will take the direction of A'C', (ZA = ZA' by hyp.). C will fall upon A'C' or its prolongation. .. the point C falling upon both the lines B'C' and A'C', must fall upon the point common to both lines, namely, C'. .. A ABC and A'B'C' coincide .. A ABC=▲ A'B'C'. Q.E.D. 69. NOTE.. This method of proof (superposition) is employed in fundamental propositions only. The student should place those parts upon each other whose equality is known, and, by successive steps, trace the position of the rest of the figure. 70. NOTE. In order to facilitate the citing of propositions, the following abbreviation is suggested for the above proposition: a. s. a. = a.s.a. Similar abbreviations will be suggested for other propositions. 71. DEF. Polygons are mutually equiangular if their angles are respectively equal, and mutually equilateral if their sides are respectively equal. If two polygons are mutually equiangular, lines or angles. similarly situated are called homologous lines or angles. Thus AB and A'B' (Prop. II) are homologous sides, C and C' homologous angles, the medians drawn from A and A' respectively homologous medians, etc. B Ex. 34. If a diagonal of a quadrilateral bisects those angles whose vertices it joins, the diagonal divides the figure into two equal triangles. PROPOSITION III. THEOREM A Ꭰ 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.), B B' Α' Hyp. In ▲ ABC and A'B'C', To prove AB = A'B', BC B'C', and B = LB'. = ▲ ABC = ▲ A'B'C'. Proof. Apply ▲ ABC to ▲ A'B'C' so that BC shall coincide 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 = 2 C. PROPOSITION IV. THEOREM 75. An exterior angle of a triangle is greater than either remote interior angle. Hyp. To prove BCD is an ext. Z of ▲ ABC. ▲ BCD > < A or ≤ B. Proof. Let E be the midpoint of BC. Draw AE and produce it its own length to F. Draw FC. In A ABE and FCE, AE= EF and BE = EC. (Con.) (Ax. 9) By joining the midpoint of AC to B, it follows in the same manner that |