CHAPTER VII. TRIANGLES. THEOREM XXIII. 176. Two triangles are congruent if two angles and an opposite side in the one are equal respectively to two angles and the corresponding side in the other. AA HYPOTHESIS. ABC and DFG As, with *G, PROOF. By 174, A + B + C = st. X = X D + * F + (89. If equals be taken from equals, the remainders are equal.) A ABC A DFG. (128. Triangles are congruent if two angles and the included side are equal in each.) THEOREM XXIV. 177. If two triangles have two sides of the one equal respectively to two sides of the other, and the angles opposite to one pair of equal sides equal, then the angles opposite to the other pair of equal sides are either equal or supplemental. The angles included by the equal sides must be either equal or unequal. CASE I. If they are equal, the third angles are equal. (174. The sum of the angles of a triangle is a straight angle.), CASE II. If the angles included by the equal sides are unequal, one must be the greater. HYPOTHESIS. ABC and FGH AS, with (128. Triangles are congruent if two angles and the included side are equal in each.) (126. In an isosceles triangle the angles opposite equal sides are equal.) But BDA + ☀ BDC = st. X, 178. COROLLARY I. If two triangles have two sides of the one respectively equal to two sides of the other, and the angles opposite to one pair of equal sides equal, then, if one of the angles opposite the other pair of equal sides is a right angle, or if they are oblique but not supplemental, or if the side opposite the given angle is not less than the other given side, the triangles are congruent. 179. COROLLARY II. Two right-angled triangles are congruent if the hypothenuse and one side of the one are equal respectively to the hypothenuse and one side of the other. On the Conditions of Congruence of Two Triangles. 180. A triangle has three sides and three angles. The three angles are not all independent, since, whenever two of them are given, the third may be determined by taking their sum from a straight angle. In four cases we have proved, that, if three independent parts of a triangle are given, the other parts are determined; in other words, that there is only one triangle having those parts: (124) Two sides and the angle between them. (128) Two angles and the side between them. (129) The three sides. (176) Two angles and the side opposite one of them. In the only other case, two sides and the angle opposite one of them, if the side opposite the given angle is shorter than the other given side, two different triangles may be formed, each of which will have the given parts. This is called the ambiguous case. Suppose that the side a and side c, and C, are given. If a> c, then, making the 4 C, and cutting off CB = a, taking B D as center, and describing an arc with radius equal to c, it may cut CD in two points; and the two unequal triangles ABC and A'BC will satisfy the required conditions. In these the angles opposite the side BC are supplemental, by 177. Loci. 181. The aggregate of all points and only those points which satisfy a given condition, is called the Locus of points satisfying that condition. 182. Hence, in order that an aggregate of points, L, may be properly termed the locus of a point satisfying an assigned condition, C, it is necessary and sufficient to demonstrate the following pair of inverses: (1) If a point satisfies C, it is upon L. (2) If a point is upon L, it satisfies C. We know from 18, that, instead of (1), we may prove its contranominal: (3) If a point is not upon L, it does not satisfy C. Also, that, instead of (2), we may prove the obverse of (1): — (4) If a point does not satisfy C, it is not upon L. THEOREM XXV. 183. The locus of the point to which sects from two given points are equal is the perpendicular bisector of the sect joining them. (129. Triangles with the three sides respectively equal are congruent.) :. ACP = BCP, .. CP is the perpendicular bisector of AB. 184. INVERSE. HYPOTHESIS. P, any point on CPI AB at its mid-point C. PROOF. AACP & BCP. 124. Two triangles are congruent if two sides and the included angle are equal in each.) |