192. THEOREM. Two triangles are congruent if the three sides of the one are equal, respectively, to the three sides of the other. B FIG. 77. B Given the AABC and A'B'C' with sides AB, BC, CA equal, respectively, to sides A'B', B'C', C'A'. To prove that ▲ ABC = ▲ A'B'C'. Proof. 1. Imagine A A'B'C' to slide along the line B'B until B' coincides with B; and then to rotate about B until side B'C' coincides with its equal side BC. If then the triangles are on the same side of BC, turn AA'B'C' over on BC as an axis. 2. Since side BA = side BA', and side CA = side CA', Why? circles may be described from centers B and C which will pass through A and A'. $ 25 3. .. if the figure is turned over on BC as an axis, the triangles will exchange places. Why? 193. Homologous parts of congruent figures. In congruent figures the line segments, angles, etc., which coincide when either figure is brought into coincidence with the other are called homologous parts. In congruent triangles homologous sides are opposite equal angles, and conversely. 194. Example. An ancient geometer measured the distance from a point A on land to a ship S in the following manner: Choosing a second point B visible from A and S, he located two stations P and Q by sighting along AS and BS, respectively. Then he located the lines AS' and BS', making a'a, and B' B. Finally he measured. the distance AS'. Justify the method. = A P α αν B Triangles ABS' and ABS are evidently congruent, since two angles and the included side of the one are equal respectively to two angles and the included side of the other (§ 189). Therefore, we conclude immediately that AS' = AS, since these sides are opposite the equal angles ẞ and B' in the two triangles. B 'S' FIG. 78. Exercises. 1. Show how to find the distance between two points A and B when the line of sight between A and B is obstructed and no third point can be found which is visible from both A and B. P Find points O and P which are visible, each from the other, and from A and B, respectively. B 2. Show how to find the distance between two inaccessible points, as the distance between two points A and B on the opposite side of a river from the observer. Choose two points P, Q visible from A and B and lay out a symmetric figure having PQ as an axis, or invent an original method. 3. In early days a device consisting of a vertical staff AC to which was attached a cross bar DE, that could be moved up and down on the staff, was used in measuring distances. Discover how the distance from A to an inaccessible point B could be measured with such an instrument. 'B' THE ISOSCELES TRIANGLE 195. Definitions. If no two sides of a triangle are equal, it is called a scalene triangle; if two sides are equal, it is called an isosceles triangle; and if the three sides are equal, it is called an equilateral triangle. The angle included by the equal sides of an isosceles triangle is called the vertical angle of the isosceles triangle, and the side opposite this angle is called the base. 196. THEOREM. In an isosceles triangle the angles opposite the equal sides are equal. Proof. 1. Draw AD bisecting / BAC and cutting BC in D. (being homologous parts of congruent triangles). 197. COROLLARY 1. If the sides of a triangle are equal, the angles are also equal. 198. COROLLARY 2. An isosceles triangle is symmetric with respect to the bisector of the angle included by the equal sides. 199. THEOREM. If two angles of a triangle are equal, the sides opposite those angles are equal. Proof. 1. Draw AD bisecting ▲ BAC and cutting BC in D. 200. COROLLARY. If the angles of a triangle are equal, the triangle is equilateral. Exercise. The following is a mariner's rule for estimating the distance of an object from a ship: Let the C ship be moving in the direction AB. Observe the moment the direction of the object C makes a given angle with the course AB, and again when it makes an angle twice as large. The distance BC will then be equal to AB, the distance through which the ship has moved between the two observations. Prove it. A B 201. Definitions. The three perpendiculars from the vertices of a triangle to the opposite sides (produced if necessary) are called the altitudes (Fig. 81); the three line segments joining the vertices to the middle points of the opposite sides are called the medians (Fig. 82); and the three line segments bisecting the angles and terminating in the opposite sides are called the bisectors of the angles of the triangle (Fig. 83). 4 4 4 FIG. 81. FIG. 82. FIG. 83. NOTE. It should be noted that the altitudes, medians, and bisectors of the angles of a triangle as here defined are line segments; the lengths of these segments are also referred to by the words altitude, median, and bisector. With regard to the latter term it should be noted, in addition, that the "bisector of an angle" is also used to denote a straight line not limited in length (see § 72); whereas here it is used to denote a segment associated with a triangle. The context will always indicate which meaning is to be attached to the term. 1. If the perpendicular bisector of one side of a triangle passes through the vertex of the opposite angle, the triangle is isosceles. Given the triangle ABC with DE, the perpen- To prove that ▲ ABC is isosceles. |