THEOREM.

Two planes may be made to coincide.

Remark. By virtue of this possibility all planes are said to be congruent.

11. If the line AB remain in the plane of the page, and be revolved about A as a pivot, it will generate an angle by its change of direction (§ 7). When the positive direction AB has changed to that of AC, the angle BAC (≤ BAC) will have been generated. When it shall have been still further changed to the direction AD, Z BAD will be the result. The rotation may still continue, and the positive direction may become in succes

sion, AE, AF, AG, AH, and may finally coincide with its first position, AB.

If the direction AB be positive, the direction AF is negative. When the positive direction has been changed to AC, the negative has been changed to AI. When the positive has been changed to AD, the negative has been

changed to AJ. When the positive has arrived at AF, the negative has arrived at AB. And when the positive has again come to the position AB, the negative will have returned to its corresponding position, AF.

Both the positive and the negative directions have generated angles, and each has made a complete rotation. In this rotation each has passed through every point in the plane. Since all positions are thus merely duplicated, the angle generated by the positive direction is, in general, the only one considered.

For convenience of description and measurement the complete rotation is separated into parts; generally into 360 equal parts, each being called a degree; sometimes into 400 equal parts, each being called a grade.

Degrees are again separated into 60 equal parts, each called a minute; each minute is separated into 60 equal parts, called seconds; and each second is then separated decimally, if a further division is necessary.

Grades are separated decimally.

Each system of division and subdivision has its advantages. The sexagesimal is the system in common use, and has been since a period of time long antedating the Christian era. Nearly all of the literature and instruments of observation involving the consideration of angles employ this system.

But the decimal system accords with our method of enumeration, and will at some time supplant the other. May the time soon come.

If the portion of the plane below the line FB be revolved on that line as an axis, it may be brought to coincide with that which is above the same line (§ 10); and in that position, if the line AB be rotated about A

as a pivot until it reach the direction AF, it will have generated the same angle in the two coincident figures. Hence the two angles thus generated must be equal. Each is one-half of the angle generated by a complete rotation; and may be expressed as 180 degrees (180°), or 200 grades (2005).

The dotted line in the accompanying figure is used to

indicate the rotation by marking the path of the point B during the generation of the angle of 360°, which is separated into two equal parts by the line FB.

If the line AB change its direction 90°, it will occupy the position in

dicated by AD, and the angles BAD and DAF will be equal.

If the part of the figure to the right of the line AD be revolved on AD as an axis until the revolved portion coincides with the part on

the left of AD, the Z DAB will coincide with the DAF; AB will fall in the direction AF; the Z BAJ will coincide with ZFAJ; and so must be equal to it.

Hence (..) the 180° of

change of direction from

(30)

AF to AB will be bisected by AJ. Thus we see that if

two lines intersect so as to form an angle of 90°, will be formed four angles of 90° each.

there

An angle of 90° is sometimes called a quadrant, but more frequently a right angle.

Lines at right angles with each other are said to be perpendicular (1); and either line with respect to the other is called a perpendicular.

When the sum of two angles is 90°, each is said to be the complement of the other.

When the sum of two angles is 180°, each is said to be the supplement of the other.

An angle that is less than 90° is said to be acute.

An angle greater than 90°, and less than 180°, is said to be obtuse.

Acute and obtuse angles are frequently spoken of as oblique.

B

E

A

H

FIG. 8.

Angles which occupy toward each other the position that AOB and EOF do, are said to be vertical. AOF and BOE are vertical.

NOTE. Although straight lines and planes both extend to infinity, it is only the relations between parts at a finite distance that concern us in the present work.

12. The sign of equality is (=).

In considering the aggregation of quantities, addition is indicated by (+), and subtraction by (−), (§ 2).

EQUALITY AXIOMS. (a) If two things are equal to a third thing, they are equal to each other.

(b) The whole is greater than any of its parts, and equals the sum of all its parts.

(c) If the same operation be performed upon equals, the results will be equal.

THEOREM. If two lines intersect, the vertical angles are equal.

By Equality Axiom (a),

▲ BAC + 2 CAF = ≤ CAF + ▲ FAG.

Then subtracting / CAF from each member of the equation, acting under the authority given by Equality Axiom (c), we have: Z BAC

Again,

LFAG.

Z BAC+ CAF = 180°,

▲ BAC + 2 BAG = 180°.

Q. E. D.