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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. An angle may be denoted also by a number or ble letter placed within, as angle 1, angle 2, angle a.
Fig. 9. 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. BZ
Fig. 10. 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 so as to form four equal angles, F 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,
G . as angle IKH. 33. An obtuse angle is
Fig. 12. greater than a right angle, but less than a straight angle, as GKI.
34. A reflex angle is one 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 ( is called
Fig. 13. the foot of the perpendicular DO.
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. Adjacent angles are those that have a common vertex and one common side, Bed as the angles ABC and CBD.
39. Vertical angles are those that have a common vertex, and sides that are pro- H longations of each other, as the angles EKH and GKF.
Fig. 14. 40. Complementary angles are those whose sum equals a right angle. Each is called the complement of the other. The angles LMN and NMO are complementary.
o Fig. 15.
Fig. 16. 41. Supplementary angles are those whose sum equals a straight angle. 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? (6) 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
18. If two lines, AB and CD, intersect in 0, making AOC = 60°, find the other angles.
19. In the same diagram, if AOC = m degrees, how many degrees are in DOB? in BOC?
20. If the four lines mentioned in Ex. 14 be drawn D
B so that AO is perpendicular to BO, and CO to DO, find Fig. 18. AOD (a) if BOC = 60°, (6) if BOC = m degrees.
21. What relation exists between the angles BOC and AOD in the preceding exercise ?
22. If, in the annexed diagram, AO is perpendicular to Co, and BO is perpendicular to Do, find AOD, (a) if COB = 40°, (6) if COB = mo.
23. What relation exists between AOD and BOC in the preceding exercise ?
24. What angle is formed by the bisectors of a pair of adjacent supplementary angles ?
25. Three lines meet in a point, O, forming six angles, 1, 2, 3, 4, 5, and 6. Find angle 3, if angle 1 = 80° and angle 5 = 60°.
GENERAL TERMS 42. A theorem is a geometrical truth requiring demonstration. 43. An axiom is a geometrical truth assumed as self-evident. 44. A problem is a question to be solved. 45. A proposition is a general term for a theorem or a problem.
46. A corollary is a theorem easily derived from another theorem.
47. A scholium is a remark.
48. A postulate is a construction so simple that its possibility is admitted without further demonstration.
49. The hypothesis is that which is assumed in the statement of a theorem.
50. The conclusion is that which follows from the hypothesis.
51. A proposition is the converse of another, when the hypothesis and the conclusion of the one are respectively the conclusion and the hypothesis of the other.
AXIOMS 1. Things that are equal to the same or equal things are equal to each other.
2. If equals be added to equals, the sums are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. If equals be added to unequals, the sums are unequal.
5. If equals be subtracted from unequals, or unequals from equals, the remainders are unequal.
6. Doubles of equals are equal. 7. Halves of equals are equal. 8. The whole equals the sum of its parts. 9. The whole is greater than any of its parts. 10. A straight line is the shortest distance between two points.
(For Axiom 11, see page 17 on parallel lines.)
POSTULATES 1. A straight line can be drawn between any two points. 2. A straight line can be produced indefinitely.