Shortest way between two Points. The Angle.

16. Theorem. The position of a straight line is determined by means of two points.

For, by the preceding axiom, these two points deter mine its direction.

17. Theorem. All the points which lie in the same direction from a given point are in the same straight

line.

Proof. Thus, if P and M (fig. 2) are in the same direction from A, the two straight lines AP and AM must likewise, by § 15, have the same direction, and must consequently coincide in the same straight line.

18. Axiom. A straight line is the shortest way from one point to another.

CHAPTER IV.

THE ANGLE.

19. Definitions. An Angle is formed by two lines meeting or crossing each other.

The Vertex of the angle is the point where its sides

meet.

The magnitude of the angle depends solely upon the difference of direction of its sides at the vertex.

a. The magnitude of the angle does not depend upon the length of its sides. Thus the angle formed by the two lines AB and AC (fig. 4) is not changed by shortening or lengthening either or both of these lines.

Right and Acute Angles; Complement and Supplement of an Angle.

b. The method of denoting the angle is by the three letters BAC, the letter A which is at the vertex being placed in the middle; or the letter A may be used by itself, when this can be done without confusion.

20. Definition. When one straight line meets or crosses another, so as to make the two adjacent angles equal, each of these angles is called a Right angle, and the lines are said to be perpendicular to each other.

Thus the angles ABC and ABD (fig. 5), being equal, are right angles.

21. Definitions. An Acute angle is one less than a right angle, as A (fig. 4).

An Obtuse angle is one greater than a right angle, as A (fig. 6).

22. Definitions. The Complement of an angle is the remainder, after subtracting it from a right angle.

The Supplement of an angle is the remainder, after subtracting it from two right angles.

23. Theorem. When one straight line meets or crosses another, the two adjacent angles are supplements of each other, and the vertical angles are equal to each other.

Proof. Let AB and CD (fig. 7) be the two lines. The adjacent angles APC and APD are supplements, for, if the perpendicular PM be erected, we have, by inspection,

b. In the same way, APC and BPC may be proved to be supplements of each other; and therefore the vertical

Adjacent and Vertical Angles. Sum of all the Angles about a Point.

angles APD and BPC must be equal, since they have the same supplement APC.

In the same way, it may be shown that the vertical angles APC and BPD are equal.

c. Corollary. If either of the angles APC, APD, BPC, or BPD is a right angle, the other three must also be right angles.

d. Scholium. As a straight line has two different directions exactly opposed to each other, it is not unfrequently considered as making an angle with itself equal to two right angles.

24. Corollary. If the two adjacent angles APC and APD (fig. 8) are supplements of each other, their exterior sides PC and PD must be in the same straight line.

25. Theorem. The sum of all the successive angles APB, BPC, CPD, DPE (fig. 9), formed in a plane, on the same side of a straight line AE, is equal to two right angles.

Proof. For it is equal to the sum of the two right angles APM, MPE, formed by the perpendicular PM.

26. Theorem. The sum of all the successive angles APB, BPC, CPD, DPE, and EPA (fig. 10), formed in a plane about a point, is equal to four right angles.

Proof. For it is equal to the sum of the four right angles MPN, NPM', MPN, NPM, formed by the two perpendiculars MM' and NN.

Parallel Lines cannot meet. Angles are equal whose Sides are Parallel.

CHAPTER V.

PARALLEL LINES.

27. Definition. Parallel Lines are straight lines which have the same Direction, as AB, CD (fig. 11).

28. Theorem.

Parallel lines cannot meet, however

far they are produced.

Proof. Thus the two lines AB and CD (fig. 11) cannot meet at P; for, if two straight lines are drawn through P, in the same direction, they must coincide and form one and the same straight line.

29. Theorem. Two angles, as A and B (fig. 12), are equal, when they have their sides parallel and directed the same way from the vertex.

Proof. For, as the directions of BD and BF are respectively the same as those of AC and AE, the difference of direction of BD and BF must be the same as that of AE and AC; that is, by § 19 the angle A is equal to the angle B.

30. Theorem. If two parallel lines AB, CD (fig. 13) are cut by a third straight line EF, the externalinternal angles, as EMB and END, or BMF and DNF, are equal, and the alternate-internal angles, as AMN and MND, or BMN and MNC, are also equal.

Proof. a. The external-internal angles are equal, because their sides have the same direction.

Angles made by a Line cutting Parallel Lines.

b. The alternate-internal angles are equal, as AMN and MND because AMN is, by § 23, equal to its vertical angle EMB, which has just been proved equal to MND.

31. Theorem. If two straight lines, lying in the same plane, as AB, CD (fig. 13), are cut by a third, EF, so that the angles EMB and END are equal, or AMN and MND are equal, &c.; the lines AB, CD must be parallel.

Proof. For the line, drawn through the point M parallel to CD, must make these angles equal, and must therefore coincide with AB.

32. Theorem. If two parallel lines AB, CD (fig. 13) are cut by a third straight line EF, the two interior angles on the same side, as BMN and MND, are supplements of each other.

Proof. For BMN is, by § 23, the supplement of its adjacent angle EMB, which is equal to MND.

33. Theorem. If two straight lines, lying in the same plane, as AB and CD (fig. 13), are cut by a third, EF, so that the angles BMN and MND are supplements of each other, the lines AB, CD must be parallel.

Proof. For the line, drawn through the point M parallel to CD, must make these angles supplements to each other, and must therefore coincide with AB.

34. Theorem. If a straight line is perpendicular to one of two parallels, it must also be perpendicular to the other.

Proof. Thus, if EMB (fig. 14), is a right angle, its equal END must also be a right angle.