GEOMETRY

PRELIMINARY DEFINITIONS

1. Every material object occupies a limited portion of space and is called a Physical Solid or Body.

2. The portion of space occupied by a physical solid is identical in form and in extent with that solid, and is called a Geometrical Solid.

In this work only geometrical solids are considered, and for brevity they are called simply solids.

3. Any limited portion of space is called a

Solid.

A solid has three dimensions, length, breadth, and thickness.

The drawing in the margin is represented as having three dimensions.

FIG. 1.

4. The limit of a solid, or the boundary which separates it from all surrounding space, is called a Surface.

A surface has only two dimensions, length and breadth.

A page of a book is a surface, but a leaf of a book is a solid.

5. The limit or boundary of a surface is called a Line.

A line has only one dimension, length. It has neither breadth nor thickness.

The edges of a cube are lines.

6. The limits, or extremities, of a line are called Points.

A point has position only. It has neither length, breadth, nor thickness.

The dots and lines made by a pencil or crayon are not geometrical points and lines, but are convenient representations of them.

7. Lines, surfaces, and solids are called Geometrical Magnitudes, or simply Magnitudes.

8. A line may be conceived of as generated by a point in motion. Hence a line may be considered as independent of a surface, and it may be of unlimited extent.

A surface may be conceived of as generated by a line in motion. Hence a surface may be considered as independent of a solid, and it may be of unlimited extent.

A solid may be conceived of as generated by a surface in motion. Hence a solid may be considered as independent of a material object.

LINES AND SURFACES

9. 1. Select two points upon your paper and draw several lines connecting them.

a. Which is the shortest line you have drawn? If this line is not the shortest that can be drawn between the points, what kind of a line is the shortest?

b. What other kinds of lines have you drawn besides a straight line?

2. When a carpenter places a straightedge upon a board and moves it about over the surface, what is he endeavoring to determine regarding the surface?

3. If the straightedge does not touch every point of the surface of the board to which it is applied, what has been discovered about the surface?

4. How does he know whether or not the surface is an even or a plane surface?

5. If any two points on the surface of a ball or sphere are joined by a straight line, where does the line pass?

6. How much of the surface of a perfect sphere is a plane surface?

10. A line which has the same direction throughout its whole

extent is called a Straight Line.

A straight line is also called a Right Line, or simply a Line.

In this work the term "line" means a

straight line unless otherwise specified.

11. A line no part of which is straight is called a Curved Line.

Consequently, a curved line changes its direction at every point.

12. A line made up of several straight lines which have different directions is called a Broken Line.

13. A line made up of straight and curved lines is called a Mixed Line.

Any portion of a line may be called a segment of that line.

FIG. 2.

FIG. 3.

FIG. 4.

14. A surface such that a straight line joining any two of its points lies wholly in the surface is called a Plane Surface, or a Plane.

15. A surface, no part of which is plane, is called a Curved Surface.

16. Any combination of points, lines, surfaces, or solids is called a Geometrical Figure.

A geometrical figure is ideal, but it can be represented to the eye by drawings or objects.

17. A figure formed by points and lines in the same plane is called a Plane Figure.

18. A figure formed by straight or right lines only is called a Rectilinear Figure.

19. The science which treats of points, lines, surfaces, and solids, and of the properties, construction, and measurement of geometrical figures, is called Geometry.

20. That portion of geometry which treats of plane figures is called Plane Geometry.

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, and the point where they meet is called its

vertex.

A

The lines OA and OB are the sides of the angle Bformed at the point O, 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

A

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.

1/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 0, 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.

If the sides OA and OB, Fig. 9, extend in opposite directions from the vertex O, the angle AOB is a straight angle.

A

B

FIG. 9.

A straight angle is equal to two right angles.