CHAPTER II RELATIONS OF TRIANGLES. Definitions. 86. Definition. An equilateral triangle is one in which the three sides are equal. Def. An isosceles triangle is one which has two equal sides. Def. An acute-angled triangle is one which has three acute angles. Equilateral Isosceles triangle. Def. A right-angled triangle is one which has a right angle. Def. An obtuse-angled triangle is one which has an obtuse angle. 8. Def. In a right angled triangle the side opposite the right angle is called the hypothe nuse. Right-angled Obtuse-angled 88. Def. When one triangle. side of a triangle has to be distinguished from the other two it is called a base, and the angle opposite the base is called the vertex. Either side of a triangle may be taken as the base, but we commonly take as the base a side which has some distinctive property. In an isosceles triangle the base is generally the side which is not equal to another. In other triangles the base is the side on which it is supposed to rest. 89. Def. An oblique line is one which is neither perpendicular nor parallel to some other line. 90. Def. Segments of a straight line are the parts into which it is divided. THEOREM IX. 91. In an isosceles triangle the angles opposite the equal sides are equal to each other. Hypothesis. ABC a triangle in which CA = CB. Conclusion. Angle A = angle B. Proof. Bisect the angle C by the line CD, meeting AB in D. Turn the triangle over on CD as an axis. and end. 1. Because, by construction, Angle BCD = ACD, Then Side CA trace CB, and vice versa. 2. Because, by hypothesis, CA = CB, Point B position A, position B. 3. Therefore line AB 4. Therefore angle CAB trace BA, being turned end for trace CBA. 5. Therefore angle CAB = angle CBA (§ 14). Q.E.D. Corollary 1. Since, after being turned over, the triangle falls upon its own trace, the triangle is symmetrical with respect to the bisecting line. Hence 92. The bisector of the vertical angle of an isosceles triangle is an axis of symmetry, and bisects the base at right angles. 93. Cor. 2. Every equilateral triangle is also equiangular. THEOREM X. 94. Conversely, if two angles of a triangle are equal, the sides opposite these angles are equal, and the triangle is isosceles. Hypothesis. ABC, a triangle in which angle A = angle B. Conclusion. Side CA side CB. Proof. Through the middle point D of the side AB pass a perpendicular, and turn the figure over upon this perpendicular as an axis. Then 1. Because the axis bisects AB perpendicularly, Point A position B.) Therefore AB= trace BA. 2. Because angle A angle B, = (§ 49) A D B 3. Therefore the point of intersection C will fall into its original position. 4. Therefore AC BC. Q.E.D. = 95. Corollary 1. A line bisecting the base of an isosceles triangle perpendicularly Passes through its vertex, Is an axis of symmetry, and Bisects the angle opposite the base. 96. Cor. 2. Every equiangular triangle is also equilateral. THEOREM XI. 97. If in any triangle one side be greater than another, the angle opposite that side will be greater than the angle opposite the other. Hypothesis. ABC, a triangle in which AB> BC > CA. Conclusion. Angle C> angle A > angle B. Proof. On a greater side, CB, take CD equal to a lesser side, A CA, and join AD. Then 1. Because the line AD falls within the triangle, Angle CAB> angle CAD. D B 2. Because CA = CD, Angle CAD angle CDA. Angle CDA > angle ABD. (§ 93) 3. Because CDA is an exterior angle of the triangle ABD, (878) 4. From (1) and (2) we have angle CAB > CDA, whence, from (3), Angle CAB > angle ABD. In the same way may be shown, Angle C> angle B. Q.E.D. THEOREM XII. 98. Conversely, if one angle of a triangle be greater than another, the side opposite the greater angle will exceed that opposite the lesser angle. Hypothesis. ABC, a triangle in which Angle C> angle A > angle B. Conclusion. Side AB > side BC> side AC. Proof. From C'draw CD, making angle A CD = angle CAD. Then 1. Because angle ACD angle CAD, = 2. Because angle ACD < ACB, the point D falls between A and B. Therefore 99. Corollary. Theorems X. and XI. may be combined into the single proposition: The order of magnitude of the three sides of a triangle is the same as the order of magnitude of their opposite angles. That is, if Angle A > angle B > angle C, then and vice versa. THEOREM XIII. 100. If from any point within a triangle lines be drawn to the ends of the base, the sum of these lines will be less than the sum of the other two sides of the triangle, but they will contain a greater angle. Hypothesis. ABC, any triangle; P, any point within it. II. Angle APB > angle ACB. Proof. Continue the line AP until it meets the side CB in Q. Then (I) 1. AQ < AC + CQ. (Ax.9) 2. Adding QB to both sides of this inequality, we have AQ+QB < AC + CQ + QB; C (II) 5. Because PQB is an exterior angle of the triangle ACQ, Angle PQB angle ACB. 6. Because APB is an exterior angle of the triangle PQB, Angle APB > angle PQB. 7. Comparing (5) and (6), Angle APB > angle ACB. Q.E.D. |