of the points in its surface. Lines, surfaces, and solids are the geometric figures. When the extent of lines, surfaces, and solids is considered they are called magnitudes, but when their form or shape is considered they are called figures.

14. Geometry is the science which treats of magnitude, form, and position. Thus it is the province of Geometry to investigate the properties of solids, of surfaces, and of the figures constructed on surfaces.

Plane Geometry treats of plane figures.

Solid Geometry, called also Geometry of Space and Geometry of Three Dimensions, treats of solids, of curved surfaces, and of the figures described on curved surfaces.

STRAIGHT LINES.

15. A finite straight line is a straight line contained between two definite points which are its extremities. When a straight line is produced indefinitely it is called an indefinite straight line. Any finite straight line may be supposed at any time to be produced into an indefinite straight line.

Two finite straight lines are said to be equal, or of equal length, when the extremities of the one line can be made to coincide respectively with the extremities of the other.

If any line, as OB, be produced

through O to A, the parts OB and OA A

are said to have opposite directions from the common point 0.

O

B

Fig. 2

Every straight line AB has two opposite directions, the one from A toward B, expressed by "the line AB,” and the other from B toward A, expressed by "the line BA."

If a line BC is to be produced toward D, we should ex

press this by saying that "BC is to A

be produced "; but if it is to be produced toward A, we should express

this by saying that "CB is to be produced."

B

D

Fig. 3

Straight lines are added together by placing them one

after another in succession in the same straight line so that one extremity of each newly added line coincides with one extremity of the last added line, and so that no part of any newly added line coincides with any part of the last added line.

Thus, AB, BC, and CD, Fig. 3, are added together and form the straight line AD.

AB, BC, and CD are called the parts of AD, and AD is called the sum of AB, BC, and CD.

PLANE ANGLES.

B

16. A plane angle is the opening between two straight lines drawn from the same point. The straight lines are called the arms or sides of the angle, and the common point is called its vertex. Thus the lines OA, OB are said to contain, or include, or form the angle at O.

A

Fig. 4

When there are several angles at one point, any one of them is expressed by three letters, putting the letter at the vertex between the other two.

C

B

Thus, if the straight lines OA, OB, OC meet at the point O, the angle contained by the lines OA, OB is named the angle AOB or BOA; the angle contained by the lines OA, OC is named the angle AOC or COA; and the angle contained by OB, OC is named the angle BOC or COB.

Fig. 5

-A

When there is only one angle at a point, it may be denoted either by the single letter at that point, or by three letters as above. Thus in Fig. 4 the angle at the point O may be denoted either by the angle O or by AOB or by BOA.

17. Adjacent angles are angles which have a common vertex and one common arm, their non-coincident arms

being on opposite sides of the common arm. Thus the angles AOB and COB (Fig. 5) are adjacent angles, of which OB is the common arm.

Of the two straight lines OB, OC (Fig. 5) it is easily seen that the opening between OA and OC is greater than the opening between OA and OB. This we express by saying that the angle AOC is greater than the angle AOB.

The magnitude or size of an angle depends entirely upon the extent of opening between its sides, and is not altered by changing the length of its sides.

18. Angles are equal when they can be placed one upon the other so that the vertex and sides of the one can be made to coincide with the vertex and sides of the other. Thus the angles ABC and DEF are equal if ABC can be placed upon DEF so that while BA coincides with ED, BC shall also coincide with EF.

B

19. The angle formed by joining two Eor more angles together is called their

Fig. 6

F

A

sum. Angles are added together by placing them so as to

Fig. 7

If the angles ABC, PQR are equal to each other, the angle ABR is double either of them, and the common side BC is said to bisect the angle between the non-coincident sides BA and BR.

20. When a straight line standing on another makes the adjacent angles equal to each other, each of the angles is

called a right angle; and the straight line which stands on the other is said to be perpendicu

lar or at right angles to it. Thus, if

the adjacent angles AOC and BOC

are equal to each other, each is a right angle, and the line CO is perpendicular to AB. The point O is called the foot of the perpendicular.

C

21. A straight angle has its arms extending in opposite directions so as to be in the same straight line. Thus, if the arms OA, OB are in the same straight. line, the angle formed by them is called a straight angle.

A

Fig. 9

Since the sum of the two right angles AOC and BOC (Fig. 8) is the angle AOB (19)*, a

right angle is half a straight angle.

22. An acute angle is an angle

which is less than a right angle, as the angle A.

Fig. 10

23. An obtuse angle is an angle which is greater than a right angle, and less than a straight

angle, as the angle BAC.

24. When the sum of two angles

is a right angle, each is called the

complement of the other, and the

two are called complementary angles. Thus, if the angle BAC is a right angle, the angles BAD, DAC are complements of each other.

25. When the sum of two angles is a straight angle, each is called the supplement of the other, and the two are called supplementary adjacent angles. Thus, if the A angle AOB (Fig. 9) is a straight angle,

Fig. 12

the angles BOC, COA are supplements of each other.

* An Arabic numeral in parenthesis refers to an article,

D

Hence, when one line stands on another, the two adjacent angles are supplements of cach other. Hence a right angle is equal to its supplement.

The supplement of an acute angle is obtuse, and, conversely, the supplement of an obtuse angle is acute.

26. A reflex angle is an angle which is greater than a straight angle, and less than two

straight angles, as the angle O.

Acute, obtuse, and reflex angles are called oblique angles, in distinction. from right and straight angles; and intersecting lines which are not per

pendicular to each other are called oblique lines.

Fig. 13

27. Where two angles are contained between two intersecting lines on opposite sides of the A vertex, they are called opposite or vertical angles. Thus, AOC and BOD are opposite or vertical angles, as also AOD and COB.

Fig. 14

ANGULAR MEASURE.

28. A right line drawn from the vertex and turning about it in the plane of the angle from the position of coincidence with one side of the angle to that of coincidence with the other side, is said to turn through the angle, and the angle is the greater as the quantity of turning is greater. Thus, suppose that the right line.

OP (Fig. 15) is capable of revolving about the point O, like the hands of a watch, but in the opposite direction, and that it has passed successively from the position OA to the positions occupied by OB, OC, OE, etc. Then it is clear that the line must have done

E

Fig. 15

B

P

-A

more turning in passing from OA to OC than in passing