7. Geometry is the science that treats of the properties of geometrical magnitudes. A 8. A straight line is a line that has the same direction throughout its length, as AB. The word "line" is frequently Eused to denote a straight line. B F FIG. 3. 9. A curved line changes its direction at every point, as CD. 10. A broken line is composed of several successive straight lines lying in different directions, as EF. 11. A plane surface or a plane is such a surface that a straight line joining any two points in the surface lies entirely in the surface. 12. A geometrical figure is any combination of solids, surfaces, lines, or points, as M or N. 13. A plane figure is a geometrical figure, all of whose points lie in the same plane, as N. 14. Plane Geometry treats of plane figures only. M N FIG. 4. 15. Solid Geometry treats of figures which are not plane. 16. When one figure can be placed upon another so that each point of the one lies upon the corresponding point of the other, the figures are said to coincide. 17. Equal magnitudes are those that can be made to coincide. 18. Proof by superposition is the method of proving the equality of two figures by means of coincidence. QUESTIONS 1. What is the path of a moving point? 2. What geometrical magnitude is, in general, generated by a moving line? by a moving surface? 3. What kind of a surface is represented by the walls of a room? LINES 19. From the definition of a straight line it appears that (a) two straight lines of unlimited length, coinciding in part, coincide throughout, (b) two straight lines can intersect only once, and (c) two points determine a straight line. The expression, straight line, is used to denote both an unlimited straight line and a part of such a line. 20. A line of definite length is also called a segment and is represented by a line whose ends are marked, as A and B (Fig. 5). 21. The length of this line is called the distance from A to B. AH C A line whose ends are not marked represents a line of indefinite length, as CD. HB D FIG. 5. 22. The direction of the line AB means the direction from A toward B; of BA, from B toward A. D А FIG. 6. 23. To produce the line AB means to prolong it through B; to produce BA means to prolong it through A. 24. To bisect a geometrical magnitude means to divide it into two equal parts. Thus, AC is bisected if AD equals A DC. D FIG. 7. 25. Two points, A and B, are equidistant from a third point, C, if CA = CB. ANGLES 26. An angle is the inclination of two intersecting lines to each other. B C FIG. 8. 27. The vertex is the point of intersection, and the lines are the sides of the angle. The lines BA and BC, meeting in B, form the angle ABC. B is the vertex, and BA and BC are the sides. 28. If there be only one angle at a vertex, it may be designated by one letter, as the angle B; but if there be two or more, three letters are necessary, as angle ABD. angle may be denoted also by a number or letter placed within, as angle 1, angle 2, angle a. An 29. To bisect an angle means to divide it into two equal parts. Thus, BC bisects the angle ABD, if angle ABC equals angle CBD. BC is called the bisector of angle ABD. 30. A straight angle is an angle whose sides lie in the same straight line but extend in opposite directions, as ACB. 31. When two straight lines intersect ro as to form four equal angles, each angle is called a right angle, as EOD, DOF, etc. A right angle is half a straight angle. 32. An acute angle is an angle less than a right angle, K E A →B H FIG. 12. an angle greater than a right angle, but less than a straight angle, as GKI. 34. A reflex angle is an angle greater than a straight angle, but less than two straight angles, as ABC. 35. Two lines are perpendicular to each other if they meet at right angles, as DN and EF (Fig. 11). The point O is called the foot of the perpendicular DO. A B FIG. 13. 36. Oblique angles are acute, obtuse, or reflex. 37. An angle is measured by finding how many times it contains a certain unit. The usual unit is the degree, or one-ninetieth () of a right angle. A degree is divided into sixty equal parts called minutes, and a minute into sixty equal parts called seconds. Degrees, minutes, and seconds are expressed by symbols, as in 6° 50' 12". Read six degrees, fifty minutes, and twelve seconds. 38. Two angles having a common vertex and a common side between them are adjacent angles, as angles ABC and CBD. 39. Vertical angles have a common vertex, and the sides of the one are pro- H longations of the sides of the other, as the angles EKH, and GKF. E -C B -D -G FIG. 14. F 40. Two angles whose sum equals a right angle are complementary angles. Each is called the complement of the other. The angles LMN and NMO are complementary. 41. Two angles whose sum equals a straight angle are supplementary angles. Each angle is called the supplement of the other. The angles PMN and NMO are supplementary. EXERCISES 4. What is the supplement of an angle of 23° ? 5. What is the complement of an angle of 45° ? 6. How many degrees are there in an angle which is twice its complement? 7. If an angle is equal to twice its supplement, what part of a straight angle is it? 8. What angle is formed by the hands of a clock at one o'clock ? at 2:30? at 2:45 ? 9. How many minutes does it take the minute hand of a clock to describe (a) a right angle? (b) an angle of 60°? (c) an angle of 45° ? 10. What is the complement of an angle of a degrees? 11. What is the supplement of an angle of n right angles ? 12. If six lines radiate from a point, forming equal angles, find one angle (a) in degrees, (b) in right angles, and (c) in straight angles. 13. What kind of an angle is less than its supplement ? equal to its supplement? greater than its supplement? 14. Four lines, AO, BO, CO, and DO, meet in a point. Which angle is the sum of AOB and BOC? the sum of BOC and COD? the difference between BOD and COD ? 15. In the same diagram, if AOB = 60°, BOC = 30°, and COD = 90°, find AOC, BOD, and AOD. 16. If the four lines meet in a point so that AOB = 120°, and BOC = COD = DOA, find BOC. 17. In Fig. 17 name four pairs of adjacent angles. 18. If two lines, AB and CD, intersect in O, making AOC = 60°, find the other angles. 19. In the same diagram, if AOC = m degrees, how many degrees are in DOB? in BOC? FIG. 17. Х 20. If the four lines mentioned in Ex. 14 be drawn D so that AO is perpendicular to BO, and CO to DO, find AOD (a) if BOC = 60°, (b) if BOC = m degrees. FIG. 18. B 21. What relation exists between the angles BOC and AOD in the preceding exercise ? |