PLANE GEOMETRY.

BOOK I

RECTILINEAR FIGURES.

ANGLES.

1. Definition. A plane angle, or simply an angle, is the amount of divergence of two lines which meet in a point or which would meet if produced (i.e., prolonged).

The point is called the vertex of the angle, and the two lines the sides of the angle.

B

From the definition it is clear that the magnitude of an angle is independent of the length of its sides. An isolated angle may be designated by the letter at its vertex, as "the angle 0;" but when several angles are formed at the same point by different lines, as OA, OB, OC, we designate the angle intended by three letters; namely, by one letter on each of its sides, together with the one at its vertex, which must be written between the other two. Thus, with these lines there are formed three different angles, which are distinguished as AOB, BOC, and AOC.

Two angles, such as AOB, BOC, which have the same vertex and a common side OB between them, are called adjacent.

2. Definitions. Two angles are equal when one can be superposed upon the other, so that the vertices shall coincide and the sides of the first shall fall along the sides of the second.

Two angles are added by placing them in the same plane with their vertices together and a side in common, care being taken that neither of the angles is superposed upon the other. The angle formed by the exterior sides of the two angles is their sum.

B

3. A clear notion of the magnitude of an angle will be obtained by supposing that one of its sides, as OB, was at first coincident with the other side OA, and that it has revolved about the point O (turning upon as the leg of a pair of dividers turns upon its hinge) until it has arrived at the position OB. During this revolution the movable side makes with the fixed side a varying angle, which increases by insensible degrees, that is, continuously; and the revolving line is said to describe, or to generate, the angle AOB. By con tinuing the revolution, an angle of any magnitude may be generated.

4. Definitions. When one straight line meets another, so as to make two adjacent angles equal, each of these angles is called a right angle; and the first line is said to be perpendicular to the second..

Thus, if AOC and BOC are equal angles,

each is a right angle, and the line CO is perpendicular to AB.

B

Intersecting lines not perpendicular are said to be oblique

to each other.

An acute angle is less than a right angle.

An obtuse angle is greater than a right angle.

5. Definition. Two straight lines lying in the same plane and forming no angle with each other—that is, two straight lines in the same plane which will not meet, however far produced—are parallel.

TRIANGLES.

6. Definitions. A plane triangle is a portion of a plane bounded by three intersecting straight lines; as ABC. The sides of the triangle are the portions of the bounding lines included between the points of intersection; viz., AB, BC, CA. The angles of the triangle are the angles formed by the sides with each other; viz., CAB, ABC, BCA. The three angular points, A, B, C, which are the vertices of the angles, are also called the vertices of the triangle.

If a side of a triangle is produced, the angle which the prolongation makes with the adjacent side is called an exterior angle; as ACD.

B

B

A triangle is called scalene (ABC) when no two of its sides are equal; isosceles (DEF) when two of its sides are equal; equilateral (GHI) when its three sides are equal.

A right triangle is one which has a right angle; as MNP, which is right-angled at N. The side MP, opposite to the right angle, is called the hypotenuse.

The base of a triangle is the side upon which it is supposed

to stand. In general, any side may be assumed as the base; but in an isosceles triangle DEF, whose sides DE and DF are equal, the third side EF is always called the base.

When any side BC of a triangle has been adopted as the base, the angle BAC opposite to it is called the vertical angle, and its angular point A the vertex of the triangle. The perpendicular AD let fall from the vertex upon

the base is then called the altitude of the triangle.

D

7. Definition. Equal figures are figures which can be made. to coincide throughout if one is properly superposed upon the other.

Roughly speaking, equal figures are figures of the same size and of the same shape; equivalent figures are of the same size but not of the same shape; and similar figures are of the same shape but not of the same size.

POSTULATES AND AXIOMS.

8. Postulate I. Through any two given points one straight line, and only one, can be drawn.

Postulate II. Through a given point one straight line, and only one, can be drawn having any given direction.

9. Axiom I. A straight line is the shortest line that can be drawn between two points.

Axiom II. Parallel lines have the same direction.

PROPOSITION I.-THEOREM.

10. At a given point in a straight line one perpendicular to the line can be drawn, and but one.

Let O be the given point in the line AB.

Suppose a line OD, constantly passing through O, to revolve about O, starting from the position OA and stopping at the position OB.

The angle which OD makes with OA will at first be less than the angle which it makes with OB, and will eventually become greater than the angle made with

OB.

B

Since the angle DOA increases continuously (3), the line OD must pass through one position in which the angles DOA and DOB are equal. Let OC be this position. Then OC is perpendicular to AB by (4).*

There can be no other perpendicular to AB at O, for if OD is revolved from the position OC by the slightest amount in either direction, one of the adjacent angles will be increased at the expense of the other, and they will cease to be equal.

11. COROLLARY. Through the vertex of any given angle one line can be drawn bisecting the angle, and but one.

Suggestion. Suppose a line OD to revolve about O, as in the proof just given.

B

* An Arabic numeral alone refers to an article in the same Book; but in referring to articles in another Book, the number of the Book is also given.