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gard a line as generated by the motion of a point, a surface as generated by the motion of a line, and a solid as generated by the motion of a surface. Again, each of these may be regarded in a purely abstract manner, distinct from each other.
Thus, we may suppose a surface to exist in space separately from the solid whose boundary it forms, and to be of unlimited extent.
Similarly, we may suppose a line to exist in space separately from the surface whose boundary it forms, and to be of unlimited length.
Likewise we may suppose a point to exist in space separately from the line, and to have only position.
The points, lines, surfaces, and solids of Geometry are called geometric points, lines, surfaces, and solids.
8. A straight line, or right line, is one which has the same direction at Aevery point, as the line AB.
9. A curved line is one no part of which is straight, but changes its C direction at every point, as the line CD.
10. A broken line is a line made up of different successive straight
E lines, as the line EF.
The word line, used alone, signifies a straight line; and the word curve, a curved line.
11. A plane surface, or, simply, a plane, is a surface in which the right line joining any two points in it lies wholly in the surface.
12. A curved surface is one no part of which is plane.
13. A figure is any definite combination of points, lines, surfaces, or solids.
A plane figure is one formed of points and lines in a plane. If the figure is formed of right lines only, it is called a rectilinear, or right-lined, figure.
The figure of a solid depends upon the relative position
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 0 to A, the parts OB and OA Aare said to have opposite directions
Fig. 2 from the common point 0.
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 express this by saying that “ BC is to
A be produced "; but if it is to be produced toward A, we should express
Fig. 3 this by saying that “CB is to be produced.”
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. 16. A plane angle is the opening between two straight lines drawn from the same point.
В The straight lines are called the arms or sides of the angle, and the common point is called its vertex. Thus the lines 0A, OB are said to contain, or
Fig. 4 include, or form the angle at 0.
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.
B contained by the lines 0A, OB is named the angle AOB or BOA; the angle contained by the lines OA, o
A OC is named the angle AOC or COA; and the angle contained by OB, OC is named the angle BOO or COB.
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 0 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 0A 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 B
A if ABC can be placed upon DEF so that while BA coincides with ED, BO shall also coincide with EF. 19. The angle formed by joining two E
Fig. 6 or more angles together is called their sum. Angles are added together by placing them so as to be adjacent to each other. Thus
R the sum of the two angles ABC, PQR, is the angle ABR, formed
A Q by applying the side QP to the
R side BC so that the vertex Q shall fall on the vertex B, and the side QR on the opposite side of BC from BA.
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 perpendicular 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 per
A pendicular to AB. The point O is
o called the foot of the perpendicular.
Fig. 8 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
-B called a straight angle.
Fig. 9 Since the sum of the two right ingles AOC and BOC (Fig. 8) is the angle AOB (19)*, a right angle i3 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
B complement of the other, and the
Fig. 11 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 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.