29. A Right Triangle is one which has a right angle; as JK L. The side opposite the right angle is called the hypothenuse. 30. An Obtuse-angled Triangle is one which has an obtuse angle; as MNO. 31. An Acute-angled Triangle is one whose angles are all acute; as DE F. Acute and obtuse-angled triangles are called oblique-angled triangles. 32. The side upon which any polygon is supposed to stand is generally called its base; but in an isosceles triangle, as DEF, in which DE = EF, the third side DF is always considered the base. THEOREM VII. 33. The sum of the angles of a triangle is equal to two right angles. Let A B C be a triangle; the sum of its three angles, A, B, C, is equal to two right angles. Produce A C, and draw CD parallel to A B; then DCE A, be- A ing external internal angles (17); B D N BCD B, being alternate internal angles (17); hence but therefore DCE+BCD+BCA=A+B+B CA DCE BCD+BCA two right angles (7); E 34. Cor. 1. If two angles of a triangle are known, the third can be found by subtracting their sum from two right angles. 35. Cor. 2. If two triangles have two angles respectively equal, the remaining angles are equal. 36. Cor. 3. In a triangle there can be but one right angle, or one obtuse angle. 37. Cor. 4. In a right triangle the sum of the two acute angles is equal to a right angle. 38. Cor. 5. Each angle of an equiangular triangle is equal to one third of two right angles, or two thirds of one right angle. 39. Cor. 6. If any side of a triangle is produced, the exterior angle is equal to the sum of the two interior and opposite. THEOREM VIII. 40. If two triangles have two sides and the included angle of the one respectively equal to two sides and the included angle of the other, the two triangles are equal in all respects. Place the side A B on its equal DE, with the point ▲ on the point D, the point B will be on the point E, as A B is equal to DE; then, as the angle A is equal to the angle D, A C will take the direction D F, and as A C is equal to D F, the point C will be on the point F; and BC will coincide with EF. Therefore the two triangles coincide, and are equal in all respects. THEOREM IX. 41. If two triangles have two angles and the included side of the one respectively equal to two angles and the included side of the other, the two triangles are equal in all respects. In the triangles ABC and DEF, let the angle A equal the angle D, the B E خ C D ABC is equal in all respects to the triangle D E F F Place the side-A C on its equal D F, with the point A on the point D, the point C will be on the point F, as A C' is equal to DF; then, as the angle A is equal to the angle D, A B will take the direction DE; and as the angle C is equal to the angle F, CB will take the direction FE; and the point B falling at once in each of the lines DE and FE must be at their point of intersection E. Therefore the two triangles coincide, and are equal in all respects. THEOREM X. 42. In an isosceles triangle the angles opposite the equal sides are equal. In the isosceles triangle ABC let AB and BC be the equal sides; then the angle A is equal to the angle C. B Bisect the angle ABC by the line BD; then the triangles ABD and A D C BCD are equal, since they have the two sides A B, B D, and the included angle ABD equal respectively to BC, B D, and the included angle D BC (40); therefore the angle A = C. 43. Cor. 1. BCD, AD From the equality of the triangles A BD and DC, and the angle ADB BDC; that is, the line that bisects the angle opposite the base of an isosceles triangle bisects the base at right angles; also, the perpendicular bisecting the base of an isosceles triangle bisects the triangle. And, conversely, the perpendicular bisecting the base of an isosceles triangle bisects the angle opposite, and also the triangle. 44. Cor. 2. An equilateral triangle is equiangular. THEOREM XI. 45. If two angles of a triangle are equal, the sides opposite are also equal. A D B In the triangle ABC, let the angle A equal the angle C; then A B is equal to BC. For if AB is not equal to BC, suppose AB to be greater than BC, and from A B cut off A D equal to BC and join DC. The triangles ADC and ABC have the two sides AD, A C, and the included angle A, respectively equal to the two sides BC, A C, and the included angle BCA; therefore the triangle A D C is equal to the triangle A B C (40), the part equal to the whole, which is absurd. In the same way it can be shown that A B is not less than BC; therefore A B is equal to BC. 46. Cor. An equiangular triangle is equilateral. THEOREM XII. 47. The greater side of a triangle is opposite the greater angle; and, conversely, the greater angle is opposite the greater side. In the triangle ABC let B be greater than C; then the side AC is greater than A B. At the point B make the angle CBD equal to the angle C; then (45) A B Conversely. Let AC AB; then the angle ABC > C. C For if the angle ABC is not greater than the angle C, it must be either equal to it or less. It cannot be equal, because then the side A B: =AC (45), which is contrary to the hypothesis. It cannot be less, because then, by the former part of this theorem, ABA C, which is contrary to the hypothesis. Hence, the angle ABC C. THEOREM XIII. 48. Two triangles mutually equilateral are equal in all respects. Let the triangle ABC have AB, BC, CA respec- tively equal to AD, DC, CA of the triangle ADC; then ABC is equal in all respects to ADC. Place the triangle ADC so that the base AC will co = D incide with its equal AC, but so that the vertex D will be on the side of AC, opposite to B. Join B D. Since by hypothesis ABAD, ABD is an isosceles triangle; and the angle ABD ADB (42); also, since BC CD, BCD is an isosceles triangle; and the angle DBCCDB; therefore the whole angle ABCA DC; therefore the triangles ABC and AD C, having two sides and the included angle of the one equal to two sides and the included angle of the other, are equal (40). |