said to describe, or to generate, the angle AOB. By continuing the revolution, an angle of any magnitude may be generated.

It is evident from this mode of generation, as well as from the definition (6), that the magnitude of an angle is independent of the length of its sides.

PERPENDICULARS AND OBLIQUE LINES.

8. Definition. 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.

Intersecting lines not perpendicular are said to be oblique to each other.

PROPOSITION I.-THEOREM.

B

C

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

B

C

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. In any one of its successive positions, it makes two different angles with the line AB; one, AOD, with the portion OA; and another, BOD, with the portion OB. As it revolves from the position OA around to the position OB, the angle AOD will continuously increase, and the angle BOD will continuously decrease. There will therefore be one position, as OC, where the two angles become equal; and there can evidently be but one.

10. Corollary. All right angles are equal. That is, the right angles AOC, BOC made by a line CO

meeting AB, are each equal to each of the right anglesA' O'C', B'O'C', made by a line C'O' meeting any other line A'B'. For, the line A'O'B' can be applied to the line AOB, so that O' shall

C

B

BI

ΟΙ

A

fall upon O, and then O'C'' will fall upon OC, unless there can be two perpendiculars to AB at O, which by the preceding proposition is impossible. The lines will therefore coincide and the angles will be equal (6).

PROPOSITION II.-THEOREM.

11. The two adjacent angles which one straight line makes with another are together equal to two right angles.

If the two angles are equal, they are right angles by the definition (8), and no proof is necessary.

C

If they are not equal, as AOD and BOD, still the sum of AOD and BOD is equal to two right angles. For, let OC be drawn at O perpendicular to AB. The angle AOD is the sum of the two angles AOC and COD. Adding the angle BOD, the sum of the two angles AOD and BOD is the sum of the three angles AOC, COD and BOD. The first of these three is a right angle, and the other two are together equal to the right angle BOC; hence the sum of the angles AOD and BOD is equal to two right angles.

B

A

12. Corollary I. If one of the two adjacent angles which one line makes with another is a right angle, the other is also a right angle.

13. Corollary II. If a line CD is perpendicular to another line AB, then, reciprocally, the line AB is perpendicular to CD. For, CO being perpendicular to AB at O, AOC is a right angle, hence (Cor. I.) AOD is a right angle, and AO or AB is perpendicular to CD.

B

C

A

14. Corollary III. The sum of all the consecutive angles, AOB, BOC, COD, DOE, formed on the same side

of a straight line AE, at a common point 0, is equal to two right angles. For, their sum is equal to the sum of the two adjacent angles AOB, BOE, which by the proposition is equal to two right angles.

B.

C

D

K

A

C

B

15. Corollary IV. The sum of all the consecutive angles AOB, BOC, COD, DOE, EOA, formed about a point O, is equal to four right angles. For, if two straight lines are drawn through 0, perpendicular to each other, the sum of all the consecutive angles formed about O will be equal to the four right angles formed by the perpendiculars.

D

C

E

16. Scholium. A straight line revolving from the position OA around to the position OB describes the two right angles AOC and COB; hence OA and OB, regarded as two different lines having opposite directions (4), are frequently said to make an angle with each other equal to two right angles.

B

D

B

C

A line revolving from the position OA from right to left, that is, successively into the positions OC, OB, OD, when it has arrived at the position OD will have described an angle greater than two right angles. On the other hand, if the position OD is reached by revolving from left to right, that is, successively into the positions OE, OD, then the angle AOD is less than two right angles. Thus, any two straight lines drawn from a common point make two different angles with each other, one less and the other greater than two right angles. Hereafter the angle which is less than two right angles will be understood, unless otherwise expressly stated.

17. Definitions. An acute angle is an angle less than a right angle; as AOD. An obtuse angle is an angle greater than a right angle; as BOD.

B

E

18. When the sum of two angles is equal to a right angle, each is called the complement of the other. Thus DOC is the complement of AOD, and AOD is the complement of DOC. 19. When the sum of two angles is equal to two right angles, each is called the supplement of the other. Thus BOD is the supplement of AOD, and AOD is the supplement of BOD.

20. It is evident that the complements of equal angles are equal to each other; and also that the supplements of equal angles are equal to each other.

PROPOSITION III.-THEOREM.

21. Conversely, if the sum of two adjacent angles is equal to two right angles, their exterior sides are in the same straight line.

Let the sum of the adjacent angles AOD,

BOD, be equal to two right angles; then, OA and OB are in the same straight line.

B

D

A

For BOD is the supplement of AOD (19), and is therefore identical with the angle which OD makes with the prolongation of AO (11). Therefore OB and the prolongation of AO are the same line.

22. Every proposition consists of an hypothesis and a conclusion. The converse of a proposition is a second proposition of which the hypothesis and conclusion are respectively the conclusion and hypothesis of the first. For example, Proposition II. may be enunciated thus:

Hypothesis-if two adjacent angles have their exterior sides in the same straight line, then-Conclusion-the sum of these adjacent angles is equal to two right angles.

And Proposition III. may be enunciated thus:

Hypothesis-if the sum of two adjacent angles is equal to two right angles, then-Conclusion-these adjacent angles have their exterior sides in the same straight line.

Each of these propositions is therefore the converse of the other. A proposition and its converse are however not always both true.

PROPOSITION IV.-THEOREM.

23. If two straight lines intersect each other, the opposite (or vertical) angles are equal.

Let AB and CD intersect in O; then will the opposite, or vertical, angles AOC and BOD be equal. For, each of these angles is the supplement of the same angle BOC, or AOD, and hence they are equal (20).

B

D

In like manner it is proved that the opposite angles AOD and

BOC are equal.

H

24. Corollary I. The straight line EOF which bisects the angle AOC also bisects its vertical angle BOD. For, the angle FOD is equal to its vertical angle EOC, and FOB is equal to its vertical angle EOA; therefore if EOC and EOA are equal, FOD and FOB are equal.

25. Corollary II. The two straight lines EOF, HOG, which bisect the two pairs of vertical angles, are perpendicular to each other. For, HOC HOB and COE = BOF; hence, by addition, HOC+ COE

=

B

F.

E

D

A

HOB

BOF; that

is, HOE HOF; therefore, by the definition (8), HO is perpendicular to FE.

PROPOSITION V.-THEOREM.

26. From a given point without a straight line, one perpendicular can be drawn to that line, and but one.

P

PI

D

B

E

Let AB be the given straight line and P the given point. The line AB divides the plane in which it is situated into two portions. Let the portion containing P, which we suppose to be the upper portion, be revolved about the line AB (i.e., folded over) until the point P comes into the lower portion; and let P' be that point in the plane with which P coincides after this revolution. Restoring P to its original position, join PP', cutting AB in C, and again revolve the upper portion of the plane about AB until P again coincides with P'. Since the line AB is fixed during the revolution, the point Cis fixed; therefore PC will coincide with P'C, and the angle PCD with the angle P'CD. These angles are therefore equal (6), and BC is perpendicular to PP' (8), or PC perpendicular to AB (13). There can therefore be one perpendicular from the point P to the line AB.

Moreover, PC is the only perpendicular. Let PD be any other