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 'D E, with the point A 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. PLANE GEOMETRY. 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 C D A B C is equal in all respects to the triangle D E F. E 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 D E 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. A B D C and Bisect the angle ABC by the line BD; then the triangles A B D and BCD are equal, since they have the two sides A B, BD, the included angle ABD equal respectively to BC, B D, and the included angle DBC (40); therefore the angle A = C. 43. Cor. 1. BCD, AD From the equality of the triangles ABD and 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. 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 A triangles ADC and ABC have the two sides D B 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 ABC (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 A C>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 A C (45), which is contrary to the hypothesis. It cannot be less, because then, by the former part of this theorem, A BAC, 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 respectively equal to AD, DC, CA of the triangle ADC; then ABC is equal in all respects to A D C. Place the triangle AD C so that the base AC will co = D incide with its equal A C, but so that the vertex D will be on the side of AC, opposite to B. Join BD. Since by hypothesis A B = AD, 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 ABC = ADC; therefore the triangles A B C 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). ·49. Scholium. In equal triangles the equal angles are opposite the equal sides. THEOREM XIV. 50. Two right triangles having the hypothenuse and a side of the one respectively equal to the hypothenuse and a side of the other are equal in all respects. A B C D Let ABC have the hypothenuse A B and the side BC equal to the hypothenuse BD and the side BC of BCD; then are the two triangles equal in all respects. Place the triangle BCD so that the side BC will coincide with its equal B C, then C D will be in the same straight line with AC (10). An isosceles triangle A B D is thus formed, and BC being perpendicular to the base divides the triangle into the two equal triangles ABC and BCD (43). THEOREM XV. 51. If from a point without a straight line a perpendicular and oblique lines be drawn to this line, 1st. The perpendicular is shorter than any oblique line. 2d. Any two oblique lines equally distant from the perpendicular are equal. 3d. Of two oblique lines the more remote is the greater. Let A be the given point, BC the given line, A D the perpendicular, and A E, AB, A C oblique lines. 1st. In the triangle A D E, the an B. E gle A D E being a right angle is greater than the angle A E D; therefore AD < A E (47). |