DEFINITIONS CONCERNING ANGLES

5. An angle is the figure formed by two indefinite half lines issuing from the same point. This point is called the vertex of the angle, and the half lines are called its sides. An angle is usually designated by three letters, the middle. one being placed at the vertex and the other two on the sides; thus the angle of the straight lines AB, AC is called the angle BAC. When, however, there is no other angle having the same vertex, the letter at the vertex is a sufficient designation. B

Α

6. A useful notion of an angle may be obtained by the conception of a revolving line.

B

In the angle AOB, imagine a line at first to coincide with OA and then to revolve about the point o (that is, to coincide in succession with different lines passing through 0) until it arrives in the position OB. The revolving line is then said to have turned through the angle AOB.

A

As there are two ways of turning a line from the position OA to the position OB, there are two angles 4OB formed by the same two half lines. These are said to be conjunct angles.

7. Two angles are said to be equal (in accordance with the definition of equal figures in general) when either angle may be transferred so as to coincide with the other, i.e. so that their sides may be coincident, and so that the two angles in question can then be turned through at the same time by the revolving half line.

When two angles are in coincidence, their conjunct angles are also in coincidence.

8. Two angles that have the same vertex with one side common, and are situated at opposite sides of this common

line, are called adjacent angles, and the whole angle formed by the two extreme lines, of which these two angles are parts, is called the

sum of the two adjacent angles. Thus,

the sum of the angles AOB and BOC is the angle AOC.

B

E D

B

Α

9. Several angles are said to be adjacent in succession when they have a common vertex and are such that the second is adjacent with the first, the third with the second, and so on without overlapping; and the whole angle formed by the two extreme lines, of which these angles are parts, is called the sum of the several angles.

For

A

instance, the angle AOE is the sum of the angles AOB, BOC, COD, and DOE. It is likewise the sum of the angles AOC, COD, and DOE.

10. Again, the sum of several angles not adjacent in succession is the angle obtained by transferring them so as to be adjacent in succession, and then taking their sum according to the preceding definition. This process is called the

addition of angles, and the given angles are then said to be added or summed. It may be stated as the process of letting a revolving line turn successively through angles equal to the given angles (such as 1, 2, 3); the whole angle thus turned through being the required sum of the separate angles.

Comparison of angles.

11. Two angles are compared in regard to magnitude by transferring one or both so that they may have the same vertex, a common side, and lie at the same side of this common line. If the other two sides happen to coincide, the angles are equal. If these sides do not coincide, one of the given angles is equal to the sum of the other given angle and a third angle. The first given angle is then said to be greater than the other, and the latter is said to be less than the former (Introd. 35).

12. The third angle, mentioned above (11), which when added to the less produces the greater, is called the difference of the two given angles. It is called also the excess of the greater over the less, or the remainder obtained by taking the less away from the greater.

13. One angle is said to be the double of another, if it is the sum of two angles each equal to the other; and the latter angle is called the half of the former.

14. It will be seen from the above definitions of the words equal, sum, difference, greater, less, double, half, when applied to angles, that in comparing the magnitude of different angles nothing is said about the magnitude of their sides. In fact, the sides of an angle may always be thought of as indefinitely prolonged.

Species of angles.

15. When the revolving half line turns from the position OA into the position 04', the prolongation of 04, it is then said to have turned through a straight angle.

Α'

Thus, an angle whose sides are in the same straight line at opposite sides of the vertex, is a straight angle.

16. If this revolving line turns through another straight angle, from O' to 04, so as to complete a revolution, the angle turned through is called a perigon.

17. The half of a straight angle is called a right angle.

18. An angle less than a right angle is called an acute angle.

19. An angle greater than a right angle and less than a straight angle is called an obtuse angle.

20. An angle less than a straight angle is called a concave angle. Two concave angles are said to be of the same species when they are both acute, both right, or both obtuse.

21. An angle greater than a straight angle and less than a perigon is called a convex angle.

22. The definitions above given and illustrated (1–21) form the basis of the statements in the next section. Further definitions will be introduced as occasion requires.

AXIOMS CONCERNING LINES AND ANGLES *

23. An axiom is a general statement whose truth ca immediately inferred from the definitions of the terms

It is convenient for purpose of reference to desig specially by the term axiom some fundamental statem relating to the equality and inequality of magnitudes, w truth can be inferred directly from the above definiti We apply these axioms only to those magnitudes which the appropriate methods of comparison have already explained.

When other kinds of magnitude are introduced, and w all the terms employed receive precise definitions as app to such magnitudes, then the appropriate axioms will stated, and their truth inferred from the definitions.

The first seven of the following axioms relate to equality of magnitudes, the remaining seven to inequal

24. Ax. 1. Magnitudes which are equal to the same n nitude are equal to each other.

25. Ax. 2. If equal magnitudes are added respectively equal magnitudes, the sums are equal.

26. Ax. 3. If equal magnitudes are subtracted respectiv from equal magnitudes, the differences are equal.

27. Ax. 4. The doubles of equal magnitudes are equal

28. Ax. 5. The halves of equal magnitudes are equal.

29. (a) Ax. 6. The sum of several magnitudes taken any order is equal to their sum taken in any other order. (b) Ax. 7. The double of the sum of two magnitudes equal to the sum of their doubles.

*The student need not dwell on Arts. 23-40 at first reading,