58 An equilateral triangle, Fig. 40, is a triangle that has its three sides equal. An equilateral triangle is a particular kind of isosceles triangle. Thus, the triangle ABC, Fig. 40, may be regarded as an isosceles triangle whose equal sides are A B and A C, as an isosceles triangle whose equal sides are BA and B C, or as an isosceles triangle whose equal sides are CA and CB. All the statements made with regard to isosceles triangles are, therefore, true of equilateral triangles. 59. Triangles are classified with regard to their angles into right-angled, obtuse-angled, and acute-angled triangles. See Fig. 41. B FIG. 40 Obtuse-angled triangle Right-angled triangle FIG. 41 Acute-angled triangle 60. A right-angled triangle, or a right triangle, is a triangle having a right angle. The hypotenuse of a right triangle is the side opposite the right angle. The legs of a right triangle are the sides that include the right angle. 61. An obtuse-angled triangle is a triangle having an obtuse angle. 62. An acute-angled triangle is a triangle all the angles of which are acute. 63. An oblique triangle is a triangle that has no right angle. The class oblique triangles includes all obtuse-angled and acute-angled triangles. 64. An equiangular triangle is a triangle whose three angles are equal. 65. The base of a triangle is the side on which the angle is supposed to stand. In a scalene triangle, any may be considered as the base. In an isosceles triangle, the unequal side is usually, though not necessarily, taken as the base. The angle opposite the base of a triangle is sometimes called the vertical angle of the triangle. In Figs. 42 and 43, AC is the base. 66. The altitude of a triangle is the length of a line drawn from the vertex of the angle opposite the base perpendicular to the base. Thus, in Figs. 42 and 43, the length of BD is the altitude. 67. An exterior angle of a triangle A is an angle formed by a side and the prolongation of another side. Thus, in FIG. 44 -+N E B D FIG. 42 TEXTECON PORERT C Figs. 43 and 44, the angle BCD, formed by the side B C and the prolongation of the side A C, is an exterior angle of the triangle ABC. The angle BCA is adjacent to the exterior angle BCD. The angles A and B are opposite-interior angles to the angle B CD. 68. In any triangle, an exterior angle is equal to the sum of the opposite-interior angles. Let DCB, Fig. 44, be an exterior angle of the triangle ABC. Draw CE through C parallel to AB. Then, the angles M and A, being exterior-interior angles, are equal. Also, N and B, being alternate-interior angles, are equal. Hence, angle M plus angle N, that is, the exterior angle DCB, is equal to angle A plus angle B, or the sum of the opposite-interior angles. 69. The sum of the interior angles of a triangle is equal to two right angles. In Fig. 44, the angles B CD and BCA, being supplementary adjacent angles, are together equal to two right angles. But, by the preceding article, the angle B CD is equal to the sum of the angles A and B. Hence, the sum of the three interior angles A, B, and B CA is equal to two right angles. 70. The following important propositions are immediate consequences of that stated in Art. 69: 1. If two angles of a triangle are known, or if their sum is known, the third angle can be found by subtracting their sum from two right angles. 2. If two angles of a triangle are equal, respectively, to two angles of another triangle, the third angle of the firstmentioned triangle is equal to the third angle of the other triangle. 3. A triangle can have but one right angle, or one obtuse angle. 4. In any right triangle, the two acute angles are complementary. 5. Each angle of an equiangular triangle is equal to onethird of two right angles, or two-thirds of one right angle. 6. From a point without a line, only one perpendicular to the line can be drawn. EXAMPLES FOR PRACTICE 1. If one acute angle of a right triangle is one-third of a right angle, what is the value of the other? Ans. Two-thirds of a right angle 2. If one angle of a triangle is one-half of a right angle, and another is five-sixths of a right angle, what is the third angle? Ans. Two-thirds of a right angle 3. The exterior angle of a triangle is 13 right angles, and one of the opposite-interior angles is one-fourth of a right angle; what are the other angles of the triangle? E Ans. Other opposite-interior angle = = * 1.15 right angles three-fifths of a right angle 4. Show that in the triangle ABC, Fig. 45, the bisector of the right angle ABC forms with the bisector of the exterior angle at C an angle that is equal to one-half of the angle A. SUGGESTION.-Let B D be the bisector of ABC and FD the bisector of BCE. Then BCF is equal to CBD plus CD B, or CDB is equal to BCF minus CBD. Also, ECB is equal to CBA plus 4, or A is equal to ECB minus CBA. Furthermore, ECB is equal to twice BCF and CBA is equal to twice C B D. 5. One angle of a triangle is one-half of a right angle: (a) What are the remaining two angles, if one is twice as large as the other? (b) What kind of triangle is this? One-half of a right angle and one right angle Ans. {(An isosceles right triangle 71. Two plane figures are equal when one can be placed on the other so that they will coincide in all their parts. Thus, the triangles A B C and A' B' C', Fig. 46, are equal, because if A'B'C' is imagined to be lifted off the paper, moved over and placed on ABC, the sides A'B', B' C', and C'A' can be made to coincide with AB, BC, and CA, respectively, and the angles A', B', and C' to coincide with the angles A, B, and C. It is evident, from the figure, that if the vertexes of the two triangles coincide, the triangles will coincide throughout, and are, therefore, equal. The polygons ABCDE and A'B'C' D' E', Fig. 47, are equal, because A'B'C' D' E' can be imagined to be lifted, turned over, and placed on ABCDE so as to make the two polygons coincide in all their parts. ᄆᄆ FIG. 47 E' 72. Two triangles are equal when a side and two adjacent angles of one are equal to a side and two adjacent angles of the other. Let A' B', Fig. 48, equal A B, the angle A' equal the angle A, and the angle B' equal the angle B. Now, if A' B' C' is placed on A B C so that A'B' coincides with its equal A B, with A' on A and B' on B, A'C' will take the direction A C; since the angle A' is equal to the angle A, and as B' is equal to B, B'C' will take the direction B C. Now, the point C will fall somewhere on the line A C, and also somewhere on the line BC, and since two lines can intersect in enly one point, C' must fall at the intersection of A Cand B C, or at C. Hence, the vertexes of the triangles coincide and the triangles are equal. The same reasoning applies to the triangles A B C and A" B" C", in which AB="A"B", and A A", B = B"; but the triangle A" B" C" and turned over before it can be placed must be imagined to be lifted on ABC. = 73. The following important principles are consequences of the preceding proposition: 1. Two triangles are equal when one side and any two angles of one are equal, respectively, to one side and the two similarly situated angles of the other. 2. Two right triangles are equal when one side and one acute angle of the one are equal, respectively, to one side and the similarly situated acute angle of the other. 74. Two triangles are equal when two sides and the included angle of one are equal to two sides and the included angle of the other. ДДД "B' FIG. 49 In Fig. 49, AB = A' B' = A" B", AC A' C A = A' = A" C", and A". If A'B'C' is placed on ABC, so that A' will coincide with A. and A'B' with AB, the rest of the triangles will evidently |