PLANE GEOMETRY

INTRODUCTION

1. The Subject Matter of Geometry.

In geometry, although we shall continue the use of arithmetic and algebra, our main work will be a study of what will later be defined (§ 13) as geometric figures. The student is already familiar with the physical objects about him, such as a ball or a block of wood. By a careful study of the following exercise, he may be led to see the relation of such physical solids to the geometric figures with which he must become familiar.

Exercise. Look at a block of wood (or a chalk box). Has it weight? color? taste? shape? size? These are called properties of the solid. What do we call such a solid? A physical

solid. Can you think of the properties of this solid apart from the block of wood? Imagine the block removed. Can you imagine the space which it occupied? What name would you give to this space? A geometric solid.

What properties has it that the block possessed? Shape and size. What is it that separates this geometric solid from surrounding space? How thick is this surface? How many surfaces has the block? Where do they intersect? How many intersections are there? How wide are the intersections? how long? What is their name? They are lines. Do these lines intersect? where? How wide are these intersections? how thick? how long? Can you say where this one is and so distinguish from where that one is? What is its name? It is a point.

If you move the block through space, what will it generate as it moves? What will the surfaces of the block generate? all of them? Can you move a surface so that it will not generate a solid? Yes, by moving it along itself. What will the edges of the block generate? Can you move an edge so that it will not generate a surface? What will the corners generate? Can you move a point so that it will not generate a line?

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FOUR FUNDAMENTAL GEOMETRIC CONCEPTS

2. The space in which we live, although boundless and unlimited in extent, may be thought of as divided into parts. A physical solid occupies a limited portion of space. The portion of space occupied by a physical solid is called a geometric solid.

3. A geometric solid has length, breadth, and thickness. It may also be divided into parts. The boundary of a solid is called a surface.

4. A surface is no part of a solid. It has length and breadth, but no thickness. It may also be divided into parts. The boundary of a surface is called a line.

5. A line is no part of a surface. It has length only. It may also be divided into parts. The boundary or extremity · of a line is called a point.

A point is no part of a line. It has neither length, nor breadth, nor thickness. It cannot be divided into parts. It is position only.

THE FOUR CONCEPTS IN REVERSE ORDER

6. As we have considered geometric solid independently of surface, line, and point, so we may consider point independently, and from it build up to the solid.

A small dot made with a sharp pencil on a sheet of paper represents approximately a geometric point.

7. If a point is allowed to move in space, the path in which it moves will be a line.

A piece of fine wire, or a line drawn on paper with a sharp pencil, represents approximately a geometric line. This, however fine it may be, has some thickness and is not therefore an ideal, or geometric, line.

8. If a line is allowed to move in space, its path in general will be a surface.

9. If a surface is allowed to move in space, its path in general will be a geometric solid.

10. A solid has threefold extent and so is said to have three dimensions; a surface has twofold extent and is said to have two dimensions; a line has onefold extent or one dimension; a point has no extent and has therefore no dimensions.

11. The following may be used as working definitions of these four fundamental concepts:

A geometric solid is a limited portion of space.

A surface is that which bounds a solid or separates it from an adjoining solid or from the surrounding space.

A line is that which has length only.

A point is position only.

DEFINITIONS AND ASSUMPTIONS

12. The primary object of elementary geometry is to determine, by a definite process of reasoning that will be introduced and developed later, the properties of geometric figures. In all logical arguments of this kind, just as in a debate, certain fundamental principles are agreed upon at the outset, and upon these as a foundation the argument is built. In elementary geometry these fundamental principles are called definitions and assumptions.

The assumptions here mentioned are divided into two classes, axioms and postulates. These, as well as the definitions, will be given throughout the book as occasion for them arises.

13. Def. A geometric figure is a point, line, surface, or solid, or a combination of any or all of these.

14. Def. Geometry is the science which treats of the properties of geometric figures.

15. Def. A postulate may be defined as the assumption of the possibility of performing a certain geometric operation. Before giving the next definition, it will be necessary to introduce a postulate.

16. Transference postulate. Any geometric figure may be moved from one position to another without change of size or shape.

17. Def. Two geometric figures are said to coincide if, when either is placed upon the other, each point of one lies upon some point of the other.

18. Def. Two geometric figures are equal if they can be made to coincide.

19. Def. The process of placing one figure upon another that the two shall coincide is called superposition.

SO

This is an imaginary operation, no actual movement taking place.

LINES

B

20. A line is usually designated by two capital letters, as line AB. It may be designated A also by a small letter placed somewhere on the line, as line a.

a

FIG. 1.

21. Straight Lines. In § 7 we learned that a piece of fine wire or a line drawn on a sheet of paper represented approximately a geometric line. So also a geometric straight line may be represented approximately by a string stretched taut between two points, or by the line made by placing a ruler (also called a straightedge) on a flat surface and drawing a sharp pencil along its edge.

22. Questions. How does a gardener test the straightness of the edge of a flower bed? How does he get his plants set out in straight rows? How could you test the straightness of a wire? Can you think of a wire not straight, but of such shape that you could cut out a piece of it and slip it along the wire so that it would always fit? If you reversed this piece, so that its ends changed places, would it still fit along the entire length of the wire? If you turned it over, would it fit? Would the piece cut out fit under these various conditions if the wire were straight?

23. Def. A straight line is a line such that, if any portion of it is placed with its ends in the line, the entire portion so placed will lie in the line, however it may be applied.

Thus, if AB is a straight line, and if any portion of AB, as CD, is placed on any other part

of AB, with its ends in AB, every point of CD will lie in AB.

C

FIG. 2.

D

B

A straight line is called also a right line. The word line, unqualified, is understood to mean straight line.

24. Straight line postulate. A straight line may be drawn from any one point to any other.

25. Draw a straight line AB. Can you draw a second straight line from A to B? If so, where will every point of the second line lie (§ 23)? It then follows that:

Only one straight line can be drawn between two points; i.e. a straight line is determined by two points.

26. Draw two straight lines AB and CD intersecting in point Show that AB and CD cannot have a second point in common (§ 23). It then follows that:

P.

Two intersecting straight lines can have only one point in common; i.e. two intersecting straight lines determine a point.

27. Def. A limited portion of a straight line is called a line segment, or simply a line, or a segment. Thus, in Fig. 2, AC, CD, and DB are line segments.

28. Def. Two line segments which lie in the same straight line are said to be collinear segments.