Similarly general and particular enunciations are commonly given for problems.

Examples of Particular Enunciation:

1. Let AB and CD be two st. lines cutting at E.

14. In general, the enunciation of a theorem consists of two parts: the hypothesis and the conclusion.

The hypothesis is the formal statement of the conditions that are supposed to exist, e.g., in the first example of § 12, "If two straight lines cut each other."

The conclusion is that which is asserted to follow necessarily from the hypothesis, e.g., "the vertically opposite angles are equal to each other."

Commonly, the hypothesis of a theorem is stated first, introduced by the word "if," and the two parts hypothesis and conclusion are separated by a comma, Sometimes, however, the two parts are not so formally

distinguished, e.g., in the proposition:-The angles at the base of an isosceles triangle are equal to each other. In order to show the two parts, this statement may be changed as follows:-If a triangle has two sides equal to each other, the angles opposite these equal sides (or angles at the base) are equal to each other.

15. The demonstration of a theorem depends either on definitions and axioms, or on other theorems that have been previously shown to be true.

The following are some of the axioms commonly used in geometrical reasoning:

1. Things that are equal to the same thing are equal to each other.

If A

= B, B = C, C = D, D = E and E = F, what about A and F?

2. If equals be added to equals the sums are equal.

A

B

Thus if A, B, C, D be four st. lines such that A = B and C D, then the sum of A and C the sum of B and D.

=

Exercise:-Mark four successive points A, B, C, D on a st. line such that AB = CD. Show that AC

3. If equals be taken from mainders are equal.

=

BD.

equals the re

Give example.

Exercise:-Mark four successive points A, B, C, D on a st. line such that AC BD. Show that AB = CD.

=

4. If equals be added to unequals the sums are unequal, the greater sum being obtained from the greater unequal.

Give example. Show also, by example, that if unequals be added to unequals the sums may be either equal or unequal.

5. If equals be taken from unequals the remainders are unequal, the greater remainder being obtained from the greater unequal.

6. Doubles of the same thing, or of equal things, are equal to each other.

7. Halves of the same thing, or of equal things, are equal to each other.

8. The whole is greater than its part, and equal to the sum of all its parts.

Give examples.

9. Magnitudes that coincide with each other, are equal to each other.

These simple propositions, and others that are also plainly true, may be freely used in proving theorems.

ANGLES AND TRIANGLES

16. Definitions. When two straight lines are drawn from a point they are said to form an angle.

A

B

The point from which the two lines are drawn is called the vertex of the angle.

The two lines are called the arms of the angle.

The angle in the figure may be called the angle BAC, or the angle CAB. The letter at the vertex

must be the middle one in reading the angle.

The single letter at the vertex is sometimes used to denote the angle when there can be no doubt as to which angle is meant.

17. Suppose a straight line OB to be fixed, like a rigid rod on a pivot at the point O, and be free to rotate in the plane of the paper.

If the line OB start from any position OA, it may rotate in either of two directions-that in which the hands of a clock rotate, or in the opposite.

When OB starts from OA and stops at any position an angle is formed with O for its vertex and OA and OB for its arms.

18. An angle is said to be positive or negative according to the direction in which the line that traces out the angle is supposed to have rotated. The direction contrary to that in which the hands of a clock rotate is commonly taken as positive.

19. The magnitude of an angle depends altogether on the amount of rotation, and is quite independent of the lengths of its arms.

20. If we wish to compare two angles ABC and DEF we may suppose the angle ABC to be placed on

C

A Ē

the angle DEF so that B falls on E and BA along ED. The position of BC with respect to EF will then show which of the angles is the greater and by how much it is greater than the other.

21. Definition.—When a revolving line OB has made half of a complete revolution from the initial position OA the angle formed is a straight angle.

The arms of a straight angle are thus in the same straight line and extend in opposite directions from