two fixed points A and B, we shall usually speak of it as the line-segment AB.

The distinction thus made between the straight line and the line-segment is sometimes effected by calling the former an unlimited straight line, and the latter a terminated straight line. Of course any mark that we make to represent a straight line must be terminated both ways by points; but the term 'line-segment' will be used only when we wish to restrict our thought to that portion of the whole line which lies between two particular points.

6. When two line-segments having one end-point in common lie in the same straight line and do not overlap, for

A

C

B

example, the segments AC and

CB, then the segment which is made up of these two, i.e. the segment AB, is called their sum.

If the line-segments have one end-point in common and do overlap, then that segment of the one which is not covered by the other is called their difference.

Thus the difference of the segments AB and CB, in the last diagram, is the segment AC.

7. A surface extends in two ways, i.e. it has length and breadth, but no thickness.

We speak of the surface of the blackboard, the surface of the table, the surface of the earth, and think of their extension in two ways only, not at all of depth or thickness. A geometrical surface has no thickness.

Any number of lines can be drawn on a surface, and any number of points can be chosen on a surface.

If a surface is such that every straight line joining two points of it lies wholly on the surface, it is called a plane surface.

The top of a table is approximately a plane surface, while the surface of an apple is clearly not plane.

For brevity we sometimes use the term a plane instead of a plane surface.

8. A solid extends in three ways, i.e. it has length, breadth, and thickness.

9. A geometrical figure is any combination of points, lines, and surfaces.

A plane geometrical figure is one whose points and lines all lie in one plane.

TRIANGLES AND ANGLES

10. One of the simplest geometrical figures is what is called a triangle. It consists of three straight lines which intersect two and two in three points.

The three points are called the vertices of the triangle, and the three straight lines its sides.

In most cases when we speak of a triangle we shall have in mind only the figure consisting of the three vertices and the three line-segments lying between them, and then the three line-segments are thought of as the sides of the triangle. It should be remembered, however, that these sides may be extended as far as we wish at any time.

The length of a side will always mean the length of the linesegment between the vertices.

A triangle may be designated by naming its vertices since no different triangle can have the same three vertices.

Thus the triangle in the diagram is the triangle ABC, or the triangle ВАС.

A

B

EXERCISE. Name the triangle in the diagram in six different ways.

Sometimes it is found convenient to designate a straight line by a single letter, and in this case we commonly use a small letter, reserving the capital letters to denote points. When the sides of a triangle are so marked we can designate the triangle by naming its three sides.

a

11. In looking at the figure of a triangle it will be noticed that the two sides which meet in any vertex start out from the vertex in different directions.

Whenever two straight lines which meet diverge from their common point in this way, they are said to form a plane angle, or simply an angle, with each other.

The idea of an angle will be made clearer by the accompanying figure. The straight lines AB and AC meet at the point A. They make with each other an angle at this point. This angle we name the angle BAC, or the angle CAB, meaning by that the angle formed by the lines BA and AC. It is sometimes spoken of as "the angle between the lines AB and AC," or "the angle contained by the lines AB and AC."

Α

If we keep the line AB fixed in position and rotate the line

AC about the point A in the way indicated by the arrowhead, we enlarge the angle BAC; that is, we increase the divergence between the lines AB and AC, and so enlarge the angle. If, however, we rotate the line the other way, the divergence becomes less; that is, the angle formed by the lines becomes smaller.

The size of the angle formed by two lines does not depend on the length of the lines, or on anything except the amount of their divergence.

The straight lines AB and AC drawn from A are called the boundaries of the angle, and their point of intersection A, the vertex of the angle.

It will readily be seen that there are three angles in any triangle, namely, the angle made by the sides AB and AC at A, that made by the sides BA and BC at B, and that made by the sides CA and CB at C. For brevity we shall sometimes speak of these as the angle A, or the angle B, or the angle C, but it must always be borne in mind that we mean the angle at A, or B, or C, made by the two straight lines which meet at that point.

The pupil should carefully observe that in reading an angle the letter at the vertex is always named between the other two letters. The angle BAC and the angle CAB are the same, having the same vertex at A and the same boundaries AB and AC, while the angle ABC is different, having its vertex at B.

12. When two angles are placed so as to have the same point for vertex and one boundary in common without overlapping, the angle formed by the other two boundaries, of which these two angles are parts, is called the sum of the two angles.

If, on the other hand, the two angles have the vertex and one boundary in common and do overlap, then the angle formed by the other two boundaries is called the difference of the two given angles.

For instance, the angle BAD in the figure is the sum of the

angle BAC and the angle CAD,

while the angle BAC is the difference of the angle BAD and the angle CAD.

When two angles have a common vertex and one common boundary, without overlapping, they are called adjacent angles.

A

D

Thus in the figure the angle BAC and the angle CAD are adjacent angles.

D

13. Suppose that on a straight line AB we choose any point C, and from it draw another straight line CD (Fig. 1). This line makes two adjacent angles with AB, namely, the angle BCD and the angle DCA. If the line be drawn, as ED (Fig. 2), so as to cross AB, then it also makes with AB two adjacent angles on the other side, namely, the angle ECB and the angle ECA.

In Fig. 2 the angles whose boundaries lie in the same straight line, but which have only their vertices in common, for example, the angle ACE and the angle BCD, are called vertically opposite angles, or briefly, vertical angles.

B

FIG. 1

EXERCISE. In this figure point out and name all pairs of adjacent angles and all pairs of vertically opposite angles. How many pairs of each kind do you find?