PLANE GEOMETRY. BOOK I. POINTS, LINES, ANGLES, POLYGONS. DEFINITIONS. 30. PLANE GEOMETRY treats of figures whose elements are all in the same plane. THE POINT. B A C 31. The position of a point is determined by its distance and direction from a known point; or by its direction from two known points, provided the three points are not all in the same straight line. Thus, the position of the point C is known, if the distance from the known point A, and the direction of C from A are known; that is, if the length and direction of A C are known. The position of the point C is also known if its direction from two known points, A and B, is known, provided A, B, and are not in the same straight line; for C, being in the lines AC and B C, must be at their point of intersection. THE STRAIGHT LINE. 32. As a straight line has the same direction throughout (7),* two points, or one point and the direction, of a straight line determine its position. * The figures alone in parentheses refer to an article in the same Book; in referring to an article in another Book, the number of the Book is prefixed. For convenience, the Introductory Definitions are numbered as though a part of Book I. 33. A straight line being the shortest distance between two points (29) is considered the distance between the points. The word line, used alone hereafter, means a straight line. 34. The origin of a line is the point at which the line is supposed to begin; or from which it is produced. Thus the line A B is produced if it is extended toward C, A being considered the origin; A B C but CB is produced, if it is extended toward A, C being considered the origin. ANGLES. 35. An Angle is the difference in direction of two lines. If the lines meet, the point of meeting, B, is called the vertex; and the lines A B, B C, the sides of the angle. B A If there is but one angle, it can be designated by the letter at its vertex, as the angle B; but when a number of angles have the same vertex, each angle is designated by three letters, the middle letter showing the vertex, and the other two with the middle letter the sides; as the angle A B C. 36. If a straight line meets another so as to make the adjacent angles equal, each of these angles is a right angle; and the two lines are perpendicular to each other. Thus, ACD and DCB, being equal, are right angles, and AB and DC are perpendicular to each other. THEOREM I. 37. All right angles are equal. D B angles A FE, CH G, E F B, G H D, are equal. Place the line A B on CD; A B will coincide with CD (32); therefore if F be considered the origin of the lines FA, FB, and H, of HC, HD, the difference of direction of FA and FB is equal to the difference of direction of HC and HD; that is, the angle formed at F by FA and FB is equal to the angle formed at H by HC and HD; and as these equal angles are respectively bisected by the perpendiculars E F, GH, the angles A FE, CHG, EFB, G HD, must be equal (28). 38. Scholium. In the measurement of angles, the right angle, being an invariable quantity, is often taken as unity. 39. Corollary. From a given point in a straight line but one perpendicular can be drawn in the same plane. DEFINITIONS. 40. An Acute Angle is less than a right angle; as EC B. 41. An Obtuse Angle is greater than a Aright angle; as AC E. Acute and obtuse angles are called oblique angles. 42. The Complement of an angle is a right angle minus the given angle. Thus (Fig. in Art. 44), the complement of A CD is ACF-ACD DCF. 43. The Supplement of an angle is two right angles minus the given angle. Thus (Fig. Art. 44), the supplement of A CD is (A CFFCB) - A CD DC B. = THEOREM II. 44. The sum of all the angles formed at à point on one side of a straight line, in the same plane, is equal to two right angles. ACD+DCE+ECB=ACD+DCF+FCE+ECB ACFFCB two right angles. 45. Cor. 1. If only two angles are formed, D each is the supplement of the other. For by the theorem, ACD+DCB= two angles; therefore AC D=two right angles-DCB, or DC B two right angles-A CD. 46. Cor. 2. The sum of all the angles formed in a plane about a point is equal to four right angles. Let the angles ABD, DBE, EBF, FBG, GBA, be formed in the same plane about the point B. Produce AB; then the sum of the angles above the line AC is equal to two right angles; and also, the sum of the angles below the line A C is equal D E A B G F to two right angles (44); therefore the sum of all the angles at the point B is equal to four right angles. THEOREM III. 47. If at a point in a straight line two other straight lines upon opposite sides of it make the sum of the adjacent angles equal to two right angles, these two lines form a straight line. Let the straight line D B meet the two lines, A B, BC, so as to make ABD DBC= two right angles : then A B and B C form a straight line. A B D E For if A B and B C do not form a straight line, draw B E so that A B and B E shall form a straight line; then ABD+DBE two right angles (44); but by hypothesis, therefore ABD+DBC two right angles; DBE: = = DBC the part equal to the whole, which is absurd (26); therefore A B and B C form a straight line. THEOREM IV. 48. If two straight lines cut each other, the opposite, or vertical, angles are equal. Let the two lines, AB, CD, cut each other at E; then For AED is the supplement of both 4. AEC and DEB (45); therefore AEC DEB D E C B THEOREM V. 49. Two angles whose sides have the same or opposite directions are equal. 1st. Let BA and B C, including the angle B, have respectively the same direction as ED and EF, including the angle E; then angle Bangle E. C A F For since BA has the same direction as ED, and BC the same as EF, the difference of direction of BA and BC must be the same as the difference of direction of ED and E F; that is, E D angle Bangle E. |