45. Axioms of geometric relation.

Ax. 9. A straight line is the shortest distance between any two of its points.

Ax. 10. If two straight lines coincide in two or more points, they will coincide throughout their whole length.

Corollary. Two straight lines can intersect in only a single point.

Ax. 11. Through a given point one straight line can be drawn, and only one, which shall be parallel to a given straight line.

The Demonstration of Theorems.

46. A theorem of geometry first supposes something to be true of a figure, and then concludes, from this supposition, that something else must be true.

That which is supposed to be true of a figure is called the hypothesis.

That which is proved to follow from the hypothesis is called the conclusion.

The hypothesis is explained for demonstration by reference to a figure.

In general, the figures as drawn need not correspond to the hypothesis. On the other hand, the hypothesis applies not simply to the figure drawn, but to every possible figure fulfilling the conditions.

The drawn figure is used to assist the beginner. In the higher investigations of geometry no figures are drawn, but letters only are used to designate them, as they are supposed to be conceived in the mind of the reader.

47. Def. When two propositions are so related that the hypothesis of each is the conclusion of the other, they are said to be the converse of each other.

THEOREM I.*

48. A straight line can be bisected in only a

single point.

Here the hypothesis supposes that we take any straight line whatever and bisect it.

To enunciate the hypothesis we call A

Р

B

one end of the line A and the other end B, and the point of bisection 0.

Then the hypothesis means that the point O is equally distant from A and B.

The conclusion asserts that there is no other point than O on the line which is equally distant from A and B.

The proof is effected by showing that to suppose any other point having this property is impossible.

If there is such a point, call it P, and suppose it between A and O (because we may call either end of the line A).

Let us then suppose that PA is equal to PB.

Because P is between A and O, AP will be less than AO. Because OB is by hypothesis equal to OA, PB, which is greater than OB, will be greater than OA.

Therefore, if we suppose PA and PB equal, PA will be greater than OA and less than OA at the same time, which is absurd. Therefore there is no point on the line except O which is equally distant from the ends of the line.

THEOREM II.

49. A straight line is symmetrical_with respect to the perpendicular passing through its middle point.

Hypothesis. AB, a straight line; O, its middle point; PQ, a perpendicular passing through 0.

* These simple theorems are presented partly as exercises and explanations for the beginner, and partly as the basis of subsequent theorems. The demonstrations are not necessarily to be recited in full as given, but the student should be encouraged and assisted in stating the substance of the reasoning in his own language.

Conclusion. The line AB is symmetrical with respect to the axis PQ.

By reference to the definition of symmetry, § 32, the conclusion

OA will fall into the position OB, and vice versa. (§ 14). Because the lengths OA and OB are equal (by hypothesis), the point A will fall on B, and vice versa.

So the conclusion is proved.

Exercise for the pupil. Prove in the same way that the line AB is symmetrical with respect to the point O as a centre of symmetry (§ 34).

THEOREM III.

50. All straight angles and all right angles are equal to each other.

To prove the first part of this proposition it is sufficient

straight angles which we may call AOB and MQN.

By the definition of a straight angle the hypothesis will mean that OA and OB go out from Q in opposite directions so that AOB is a straight line, and that the same is true of QM and QN.

NOTE. The meaning of the hypothesis apart from its enunciation must always be clearly apprehended by the student.

The conclusion is that the angles AOB and MQN are equal.

By the definition of equality in angles, (§ 14), this will mean that if we apply AOB to MQN so that the side AO shall coincide with MQ, then the side OB will coincide with QN.

This must be the case, B because two straight lines. coincide throughout when any two of their points N coincide. (§ 45, Ax. 10).

Therefore the conclusion is proved, because, from the fact that any two straight angles are equal, it follows that all are equal.

Because a right angle is, by definition, the half of a straight angle, and because all straight angles are equal, it follows from § 44, Axiom 5, that all right angles are equal.

THEOREM IV.

51. The sum of all the angles formed on one side of a straight line by lines

emanating from a point on it is a straight angle.

Proof. If O be the point from which the lines emanate and OB,

OC, etc., the lines emanating

B

-E

from it, then, by definition (§ 18), the sum of all the angles, AOB, BOC, etc., to DOE, will be the angle AOE; that is, a straight angle; because AOE is a straight line.

52. Corollary. The sum of all the angles around a point is equal to two straight angles, or a circumference.

BOOK II.

FUNDAMENTAL PROPERTIES OF RECTILINEAL FIGURES.

CHAPTER I.

RELATIONS OF ANGLES.

Definitions.

53. Def. A rectilineal figure is one which is formed by straight lines.

54. Def. A triangle is a figure formed by three straight lines joined end to end.

55. Def. The three lines which form a triangle are called its sides.

56. The sides of a triangle

may be produced indefinitely.

It is then called a general triangle.

A triangle.

REMARK. Any three indefinite straight lines which intersect each other in three different points form a general triangle. See figure on following page.

57. An interior

angle of a triangle is one between two sides, measured inside the triangle.

58. An exterior angle of a triangle is

Exterior angle.

Interior angle.

one which is formed between any side and the continu

ation of another side.