21. That portion of geometry which treats of figures whose points and lines do not all lie in the same plane is called Solid Geometry.

ANGLES

22. 1. From any point draw two straight lines in different directions. Draw two straight lines from each of several other points, and thus form several angles.

2. How does the angle at the corner of this page compare in size with the angle at the corner of the room? answer to be true by an actual test.

Show your

How is the size of any angle affected by the length of the lines which form its sides?

3. Form several angles at the same point; that is, several angles having a common vertex.

4. How many of them have a common vertex and one common side between them and are, at the same time, on opposite sides of the common side; that is, how many angles are adjacent angles?

5. Draw a straight line meeting another straight line so as to form two equal adjacent angles; that is, two right angles.

6. Draw from a point or vertex two straight lines in opposite directions; that is, form a straight angle. How does a straight angle compare in size with a right angle?

7. Draw several angles, some greater and some less, than a right angle.

8. Draw a right angle and divide it into two parts, or into two complementary angles.

9. Draw a straight angle and divide it into two parts, or into two supplementary angles.

10. Draw two straight lines crossing or intersecting each other, thus forming two pairs of opposite or vertical angles.

23. The difference in direction of two lines which meet is called a Plane Angle, or simply an Angle.

The lines are called the sides of the angle, A

and the point where they meet is called its vertex.

The lines OA and OB are the sides of the angle Bformed at the point 0, and O is the vertex of the angle.

FIG. 5.

The size of an angle does not depend upon the length of its sides, but upon the divergence of the sides or upon the opening between them. Compare Figs. 5 and 6.

B

FIG. 6.

24. When there is but one angle at a point, it may be designated by the single letter at the vertex, or by three letters.

In Fig. 6 the angle may be called the angle A, or the angle BAC, or the angle CAB.

When several angles have a common vertex, it is customary to use three letters in designating each, placing the letter at the vertex between the other two.

An angle is sometimes designated by a figure or small letter placed in the opening of the angle.

A

FIG. 7.

The angles formed by the lines meeting at O may be designated by AOC, the figure 1, and the small letter a.

B

25. Angles which have a common vertex and a common side, and which are upon opposite sides of the common side, are called Adjacent Angles.

In Fig. 7 angles COA and COB are adjacent angles, having a common vertex O, and a common side CO and lying upon opposite sides of the common side. Also COB and BOD are adjacent angles.

26. When one straight line meets another straight line so as to form two equal adjacent angles, each of the angles is called a Right Angle; and each line is said to be

perpendicular to the other.

The sides of a right angle are therefore perpendicular to each other, and lines per

pendicular to each other form right angles B

with each other.

FIG. 8.

27. An angle whose sides extend in opposite directions from the vertex, thus forming one straight line, is called a Straight Angle.

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.

If the angles AOD and DOB are to

gether equal to two right angles, the

angles AOD and DOB are supplemen

FIG. 14.

B

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.

The angles AOC and DOB, and the angles

AOD and COB are vertical angles.

34. A line, or a plane, which divides

any geometrical magnitude into two

B

FIG. 15.

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 a 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.

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 04 to 04, 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 direction, 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.

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 cannot intersect in more than 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?

How does the sum of the consecutive angles formed on both sides 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?