Feb. 18, PROPOSITION XVI.—THEOREM. 60. If two angles have their sides respectively parallel and lying in the same direction, they are equal. Let the angles ABC, DEF, have their sides BA and ED parallel and in the same direction, and also their sides BC and EF parallel and in the same direction. Then ABC DEF. = For, let DE, produced if necessary, intersect BC in G. The angle DGC is equal to its corresponding angle ABC and also to its corresponding angle DEF (51); therefore ABC = DEF. = Note. Two parallels, as BA and ED, are said to be in the same direction when they lie on the same side of the indefinite straight line joining the origins, B and E, of these parallels. 61. Corollary I. Two angles, as ABC and D'EF', having their sides parallel and lying in opposite directions (that is ED' opposite to BA and EF' opposite to BC), are equal. For we have D'EF' = DEF = ABC. 62. Corollary II. Two angles, as ABC and DEF', having two of their sides, BA and ED, parallel and in the same direction, while their other two sides, BC and EF', are parallel and in opposite directions, are supplements of each other. A 63. Corollary III. If two angles, ABC, DEF, have their sides perpendicular each to each, that is, AB to ED and BC to EF, they are either equal or supplementary. For, suppose the angle DEF to be revolved into the position HEK, by revolving ED and EF each through a right angle; that is, ED through the right angle DEH and EF through the right angle FEK. Then EH D B F C H E being perpendicular to ED is parallel to AB, and EK being perpendicular to EF is parallel to BC (44); therefore HEK, or DEF, is either equal to ABC by (60) or (61), or it is the supplement of ABC by (62). TRIANGLES. A 64. Definitions. A plane triangle is a portion of a plane bounded by three intersecting straight lines; as ABC. The sides of the triangle are the portions of the bounding lines included between the points of intersection; viz., AB, BC, CA. The angles of the triangle are the angles formed by the sides with each other; viz., CAB, ABC, BCA. The three angular points, A, B, C, which are the vertices of the angles, are also called the vertices of the triangle. If a side of a triangle is produced, the angle which the prolongation makes with the adjacent side is called an exterior angle; as ACD. B A 65. A triangle is called scalene (ABC) when no two of its sides are equal; isosceles (DEF) when two of its sides are equal; equilateral (GHI) when its three sides are equal. A right triangle is one which has a right angle; as MNP, which is right-angled at N. The side MP, opposite to the right angle, is called the hypotenuse. The base of a triangle is the side upon which it is supposed to stand. In general any side may be assumed as the base; but in an isosceles triangle DEF, whose sides DE and DF are equal, the third side EF is always called the base. When any side BC of a triangle has been adopted as the base, the angle BAC opposite to it is called the vertical angle, and its angular point A the vertex of the triangle. The per B D pendicular AD let fall from the vertex upon the base is then called the altitude of the triangle. PROPOSITION XVII.-THEOREM. 66. Any side of a triangle is less than the sum of the other two. Let BC be any side of a triangle whose other two sides are AB and AC; then BC < AB + AC. For, the straight line BC is the shortest distance be- B tween the points B and C. A 67. Corollary. Any side of a triangle is greater than the difference of the other two. For, if from each member of the inequality BC < AB+ AC we subtract AB, we shall have BC-ABAC, or AC> BC — AB. PROPOSITION XVIII.-THEOREM. 68. The sum of the three angles of any triangle is equal to two right angles. Let ABC be any triangle; then, the sum of its three angles, A, B and C, is equal to two right angles. B E For, produce BC to D, and draw CE parallel to BA. The angle ACE is equal to its alternate angle BAC (49), and the angle ECD is equal to its corresponding angle ABC (51). Therefore the sum of the three angles of the triangle is equal to ECD + ACE + BCA, which is two right angles (14). 69. Corollary I. Any exterior angle, as A CD, is equal to the sum of the two opposite interior angles, A and B; and consequently greater than either of them. 70. Corollary II. If one angle of a triangle is a right angle, or an obtuse angle, each of the other two angles must be acute; that is, a triangle cannot have two right angles, or two obtuse angles. 71. Corollary III. In a right triangle, the sum of the two acute angles is equal to one right angle; that is, each acute angle is the complement of the other (18). 72. Corollary IV. If two angles of a triangle are given, or only their sum, the third angle will be found by subtracting their sum from two right angles. 73. Corollary V. If two angles of one triangle are respectively equal to two angles of another triangle, the third angle of the one is also equal to the third angle of the other. PROPOSITION XIX.-THEOREM. 74. The angle contained by two straight lines drawn from any point within a triangle to the extremities of one of the sides is greater than the angle contained by the other two sides of the triangle. From any point D, within the triangle ABC, let DB, DU be drawn; then, the angle BDC is greater than the angle BAC. For, produce BD to meet AC in E. We have the angle BDCBEC (69), and the angle BEC> BAC; hence BDC> BAC B D E 75. Definition. Equal triangles, and in general equal figures, are those which can exactly fill the same space, or which can be applied to each other so as to coincide in all their parts. PROPOSITION XX.-THEOREM. 76. Two triangles are equal when two sides and the included angle of the one are respectively equal to two sides and the included angle of the other. A D D In the triangles ABC, DEF, let AB be equal to DE, BC to EF, and the included angle B equal to the included angle E; then, the triangles are equal. For, the triangle ABC may be superposed upon B CE E the triangle DEF, by applying the angle B to the equal angle E, the side BA upon its equal ED, and the side BC upon its equal EF. The points A and C then coinciding with the points D and F, the side AC will coincide with the side DF, and the triangles will coincide in all their parts; therefore they are equal (75). 77. Corollary. If in two triangles ABC, DEF, there are given B=E, AB DE and BC EF, there will follow AD, CF, and AC DF. = = PROPOSITION XXI.-THEOREM. 78. Two triangles are equal when a side and the two adjacent angles of the one are respectively equal to a side and the two adjacent angles of the other. In the triangles ABC, DEF, let BC be equal to EF, and let the angles B and C adjacent to BC be respectively equal to the angles E and F adja cent to EF; then, the triangles are equal. For, the triangle ABC A D D may be superposed upon the triangle DEF, by applying BC to its equal EF, the point B upon E, and the point Cupon F. The angle B being equal to the angle E, the side BA will take the direction of ED, and the point A will fall somewhere in the line ED. The angle C being equal to the angle F, the side CA will take the direction of FD, and the point A will fall somewhere in the line FD. Hence the point A, falling at once in both the lines ED and FD, must fall at their intersection D. Therefore the triangles will coincide throughout, and are equal. 79. Corollary. If in two triangles ABC, DEF, there are given B = E, C F, and BC= EF, there will follow A = D, AB DE, and AC = DF. = = PROPOSITION XXII.-THEOREM. 80. Two triangles are equal when the three sides of the one are respectively equal to the three sides of the other. In the triangles ABC, DEF, let AB be equal to DE, AC to DF, and BC to EF; then, the triangles of EF from D, as at G; and join DG which intersects EF in H. |