DOE; and the angle DOE is the complement of the angle COD.

32. When two angles are together equal to two right angles,

they are called Supplementary An

gles, and each is said to be the Supplement of the other.

tary angles; the angle AOD is the supplement of the angle DOB, and the angle DOB is the supplement of the angle AOD.

33. When two lines intersect, the opposite angles are called Vertical Angles.

-D

The angles AOC and DOB, and the angles

AOD and COB are vertical angles.

34. A line or a plane which divides

B

FIG. 15.

any geometrical magnitude into two

equal parts is called the Bisector of that magnitude.

MEASUREMENT OF ANGLES

35. To measure a magnitude is to find how many times it contains á certain other magnitude assumed as a unit of measure.

The unit of measure for angles is sometimes a right angle, but

very often it is a degree.

B

A

Suppose the line OB, having one of its extremities fixed at 0, moves from a position coincident with 04 to the position indicated by OB. By this motion the angle AOB has been generated. When the rotating line OB has passed one half the distance from OA around to OA, the lines extend in opposite directions from 0, and a straight angle has been generated; and since a straight angle is equal to two right angles (§ 27), when the line has passed one fourth of the distance around to OA, a right angle has been generated, and the lines OB and 04 are perpendicular to each other (§ 26). When the line has rotated entirely around from OA to OA, it has generated two straight angles, or four right angles. Consequently: The total angular magnitude about a point in a plane is equal to four right angles.

FIG. 16.

Inasmuch as it is frequently convenient to employ a smaller unit of angular measure than a right angle, the entire angular magnitude about a point has been divided into 360 equal parts, called degrees; a degree into 60 equal parts, called minutes; a minute. into 60 equal parts, called seconds.

Degrees, minutes, and seconds are indicated in connection with. numbers by the respective symbols, ',".

25 degrees, 18 minutes, 34 seconds is written 25° 18' 34". A right angle is an angle of 90°.

EQUALITY OF GEOMETRICAL MAGNITUDES

36 Geometrical magnitudes which coincide exactly when one is placed upon or applied to the other are equal. Since, however, geometrical magnitudes are ideal they are not actually taken up and placed the one upon the other, but this is conceived to be done.

This method of establishing equality is called the Method of Superposition.

If one straight line is conceived to be placed upon another straight line so that the extremities of both coincide, the lines are equal.

If an angle is conceived to be placed upon another angle so that their vertices coincide and their sides take the same directions, respectively, the angles are equal.

If any figure is conceived to be placed upon any other figure so that they coincide exactly throughout their whole extent, they are equal.

Figures that are superposable are sometimes called congruent.

EXERCISES

37. Draw as accurately as possible the figures which are suggested; study them carefully; infer the answers to the questions; state your inference or conclusion in as accurate form as possible; give the reason for your conclusion when you can.

The student is asked to represent by a drawing any figure that may be required so that it may simply appear to the eye to be accurate. Geometrical methods of construction are given at suitable points in the book, but they cannot be insisted upon at this stage.

1. Draw two straight lines intersecting in as many points as possible. In how many points do they intersect?

Inference: Two straight lines can intersect in only one point.

2. Draw a straight line; draw another meeting it. How does the sum of the adjacent angles thus formed compare with two right angles?

Inference: When one straight line meets another straight line, the sum of the adjacent angles is equal to two right angles.

3. Draw a straight line; from any point in it draw several lines extending in different directions. How does the sum of the consecutive angles formed on one side of the given line compare with a right angle? With a straight angle?

4. Draw a straight line; also another meeting it so as to form two adjacent angles, one of which is an acute angle. What kind of an angle is the other?

5. Draw two intersecting lines. How many angles are formed? How do the opposite or vertical angles compare in size?

6. Draw two lines intersecting so as to form a right angle. How does each of the other angles formed compare with a right angle? How do right angles compare in size? How do straight angles compare in size?

7. Draw two equal angles. How do their complements compare? How do their supplements compare

8. Draw a straight line; select any point in that line and draw as many perpendiculars as possible to the line at that point. How many such perpendiculars can be drawn on one side of the line?

DEMONSTRATION OR PROOF

38. The inferences which the student has just made are probably correct, but they must be proved to be true before they can be relied upon with certainty unless their truth is self-evident.

Many truths have been inferred, and used as the basis of important enterprises before they have been logically demonstrated. Carpenters believe that their squares are true if a line from the 12-inch mark on one side to the 16-inch mark on the other is 20 inches long; but they may not be capable of giving satisfactory reasons for their convictions.

Many valuable facts of geometry may be inferred by observation of figures and objects, but the value of the study to a student consists not so much in the knowledge acquired as in the development of the logical faculty by the rigid course of reasoning required to prove the truth or falsity of the inference.

Much attention must therefore be given to the demonstration or proof of inferences from known data, and of statements even though they may seem to be true.

39. A course of reasoning which establishes the truth or falsity of a statement is called a Demonstration, or Proof.

40. A statement of something to be considered or done is called a Proposition.

"All men are mortal" and "It is required to bisect an angle sitions.

are propo

41. A proposition so elementary that its truth is self-evident is called an Axiom.

An axiom is a self-evident truth to those only who understand the terms employed in expressing it.

Axioms may be illustrated, but they do not require proof.

Axioms have often a general application. Some, however, apply only to geometrical magnitudes and relations.

"A whole is equal to the sum of all its parts" is a general axiom. It can be employed in demonstrating propositions in arithmetic and algebra as well as in geometry. "A straight line is the shortest distance between two points" is a geometrical axiom. It can be used only in proving propositions which express some geometrical truth.

42. A proposition which requires demonstration or proof is called a Theorem.

"In any proportion the product of the extremes is equal to the product of the means " is an algebraic theorem.

43. A theorem whose truth may be easily deduced from a preceding theorem is often attached to it, and called a Corollary.

The arithmetical theorem, "A number is divisible by 3 when the sum of its digits is divisible by 3" may be readily deduced from the theorem, "A number is divisible by 9 when the sum of its digits is divisible by 9,” and may be attached to it as a corollary.

44. A proposition requiring something to be done is called a Problem.

"Construct an angle equal to a given angle" is a geometrical problem.

45. A problem so simple that its solution is admitted to be possible is called a Postulate.

"A straight line may be drawn from one point to another" is a postulate. Postulates are numerous. Some of those employed in geometry may be

found in § 50.

46. A remark made upon one or more propositions, and showing, in a general way, their extension or limitations, their connection, or their use is called a Scholium.

Thus, after the processes of dividing a common fraction by a common fraction, and a decimal by a decimal, have been taught, a remark showing that precisely the same principles are involved in each process is a scholium.