PROPOSITION XIV. THEOREM 53. Two angles whose sides are perpendicular, each to each, are either equal or supplementary. GIVEN―the angle NOM, or a, and the lines AB and CD intersecting at O' and respectively perpendicular to ON and OM. TO PROVE the angle a=a', and a+b=2 right angles. At O, draw OA' parallel to AB and OC' parallel to CD. OA', being parallel to AB, is perpendicular to ON. § 36 [If two straight lines are parallel, and a third straight line is perpendicular to one of them, it is perpendicular to the other.] For the same reason OC', being parallel to CD, is perpendicular to OM. From each of the right angles A'ON and C'OM take away the common angle w. This leaves c=a. Ax. 3 851 [Having their sides respectively parallel, and in the same right-and-left But 54. Remark.-The angles are equal if their sides are perpendicular right to right and left to left, but supplementary if their sides are perpendicular in opposite right-and-left order. Thus a and DO'B, which have their right sides (OM and O'D) perpendicular and their left sides (ON and O'B) perpendicular, are equal; etc., etc. TRIANGLES 55. Def.—A triangle is a figure bounded by three straight lines called its sides. 56. Def.—A right triangle is a triangle one of whose angles is a right angle. 57. Def.-An equiangular triangle is one whose angles are all equal. 58. The sum of the three angles of any triangle is two right angles.* AV K. H GIVEN TO PROVE ABC, any triangle, with a, b, and c its angles. a+b+c=2 right angles. Draw KH parallel to BC, and from O, any point of this line, draw OE and OD parallel respectively to the sides AB and AC. *This was first proved by Pythagoras or his followers about 550 B.C. [Having their sides parallel and in the same right-and-left order.] Hence But a+b+c=a+b'+c'. a+b+c=2 right angles. Ax. 2 827 [The sum of all the angles about a point on one side of a straight line equals two right angles.] Hence a+b+c=2 right angles. Ax. I Q. E. D. 59. Cor. I. If one side of a triangle be produced, the exterior angle thus formed equals the sum of the two opposite interior angles (and hence is greater than either of them). и с x OUTLINE PROOF: a+b+c=2 right angles=x+c, whence a+b=x. 60. COR. II. If the sum of two angles of a triangle be given, the third angle may be found by taking the sum from two right angles. [What axiom applies ?] 61. COR. III. If two angles of one triangle are equal respectively to two angles of another triangle, the third angles will be equal. [What two axioms apply?] 62. COR. IV. A triangle can have but one right angle, or one obtuse angle. 63. COR. V. In a right triangle the sum of the two angles besides the right angle is equal to one right angle. 64. COR. VI. In an equiangular triangle, each angle is one-third of two right angles, and hence two-thirds of one right angle. 65. Defs.—A polygon is a figure bounded by straight lines called its sides. A polygon is convex, if no straight line can meet its sides in more than two points. 66. The sum of all the angles of any polygon is twice as many right angles as the figure has sides, less four right angles. From any point O within the polygon draw lines to all the vertices forming ʼn triangles. The sum of the angles of each triangle is equal to 2 right angles. $58 Hence the sum of the angles of the n triangles is equal to 2n right angles. But the angles of the polygon make up all the angles of all the triangles except the angles about O, which make 4 right angles. 828 Hence the sum of the angles of the polygon is 2n-4 right angles. Q. E. D. 67. Defs.—A quadrilateral is a polygon of four sides, a pentagon, of five, a hexagon, of six, an octagon, of eight, a decagon, of ten, a dodecagon, of twelve, a pentedecagon, of fifteen. 68. Exercise. The sum of the angles of a quadrilateral equals what? of a pentagon? of a hexagon? 69. If the sides of any polygon be successively produced, forming one exterior angle at each vertex, the sum of these exterior angles is four right angles. GIVEN the polygon P with successive exterior angles a, b, c, d, e. TO PROVE a+b+c+d+e=4 right angles. Through any point O draw lines successively parallel to the sides produced. [Two angles are equal if their sides are parallel and in the same order.] |