178. Equal arcs, being congruent, have equal chords. Therefore, also, explemental arcs have equal chords. 179. Equal chords, having equal minor arcs, which may be brought into coincidence by rotation about the center, bave also equal perpendiculars from the center. 180. In the same or equal circles, chords which have equal perpendiculars from the center, since by rotation one may be put upon the other, are equal. 181. Since the end point of a chord AC AB lies on the minor arc AB, .. it is on the side of AB remote from the center O; the mid point of chord AC is on the şide of AB remote from 0. FIG. 80. B Theorem. In the same or equal circles, the greater chord has the lesser perpendicular from the center. 182. Inversely. The chord with the greater perpendicular from the center is the lesser; for [181] it cannot be the greater chord, nor [179] can they be equal. 183. Problem. At a given point G, in a given straight s, to make an angle equal to a given angle ACB. K1 H2 H1C. D K3 K 4 FIG. 81. B Construction. With any radius. A draw OC [r], cutting CA at D, and CB at F. Join DF. Draw OG [r], cutting the given straight at H. Draw ©H [r' = DF], cutting OG [r] at K. Join GK. Then HGK = ACB. Proof. OG [r]= OC [r], and chord HK = chord DF. .. minor arc HK = minor arc DF. .. minor HGK = minor ¥ DCF. Determination. The construction will give four minor angles. at G, all equal, namely, H1GK1; KGH2; KGH ̧; H„GK ̧. CHAPTER VII. TWO CIRCLES. 184. A figure formed by two circles is symmetrical with regard to their center-straight as axis. о FIG. 82. Every chord perpendicular to this axis is bisected by it. If the circles have a common point on this straight, they cannot have any other point in common, for any point in each has its symmetrical point with regard to this axis, and circles with three points in common coincide. 185. Two circles with only one point in common are called tangent, are said to touch; and the common point is called the point of tangency or contact. ӨӨ FIG. 83. 186. If two circles touch, then, since there is only one common point, this point of contact lies on the center-straight, and a perpendicular to the center-straight through the point of contact is a common tangent to the two circles. CHAPTER VIII. PARALLELS. 187. A straight cutting across other straights is called a transversal. [In plane geometry, all are in one plane.] 188. If, in a plane, two straights are cut in two distinct points by a transversal, at each of these points four positive minor angles are made. 1 FIG. 84. Of these eight angles, four are between the two straights. [namely, 3, 4, a, b], and are called Interior Angles: the other four lie outside the two straights, and are called Exterior Angles. ba са FIG. 85. Angles, one at each point, which lie on the same side of the transversal, the one exterior and the other interior, are called Corresponding Angles [e.g., 1 and a]. Two non adjacent angles on opposite sides of the transversal, and both interior or both exterior, are called Alternate Angles [e.g., 3 and a]. Two angles on the same side of the transversal, and both interior or both exterior, are called Conjugate Angles [e.g., 4 and a]. 189. Theorem. If two corresponding or two alternate angles are equal, or two conjugate angles are supplemental, then every angle is equal to its corresponding and to its alternate, and supplemental to its conjugate. [Use vertical angles and supplemental adjacent angles.] FIG. 86. FIG. 87. 34 ba FIG. 88. 190. Parallels are straights in the same plane which nowhere meet. [Note. As we are working on a plane, the clause "in the same plane" would be understood even if not mentioned.] 191. ASSUMPTION V. Two coplanar straights are parallel if a transversal makes equal alternate angles. 192. ASSUMPTION VI. If two coplanar straights cut by a transversal have a pair of alternate interior angles unequal, they meet on that side of the transversal where lies the smaller angle. 193. Theorem. If two straights cut by a transversal have corresponding angles equal, or conjugate angles supplemental, they are parallel. For either hypothesis makes the alternate angles equal. 194. If two straights cut by a transversal have conjugate angles not supplemental, they meet. B For the alternate angles are unequal. C FIG. 89. 195. Problem. Through a given D point to draw a parallel to a given straight. Construction. Join the given point P to any point C, of the given straight CB. Then at P make an angle CPD alternate and equal to PCB. Determination. There is only one solution. 196. Corollary. Two coplanar straights parallel to the same straight are parallel to one another. For they cannot meet. 197. Theorem. If a transversal cuts two parallels, the alternate angles are equal. A M Proof. For if they were unequal, the straights would meet. 198. Theorem. Any two parallels c are symcentral with regard to the mid. point of the sect which they intercept on any transversal. FIG. 90. Proof. Rotating the figure about M through a straight angle brings A into coincidence with the trace of B and X CAM into coincidence with the trace of the equal alternate X DBM. 199. Two angles with their arms parallel are either equal or supplemental [189 and 197.] FIG. 91. 200. If two angles have their arms respectively perpendicular, they are either equal or supple 202. Problem. To pass a circle through any three points not costraight. Construction. Join the three points by three sects; to these sects erect r't bisectors; of these every an angle or arms they are two will meet, since they make sects. FIG. 93. Therefore it is the center of a circle containing the three given points. 203. Corollary. The center of any through the three points must lie on all three r't bi's. |