THEOREMS FOR EXERCISE.

14. If from a point within a triangle two straight lines be drawn to the extremities of the base, these straight lines shall be together less than the two sides of the triangle.

15. Two villages, A and B, are connected by straight roads with two others, C and D. The roads AD and CB intersect, but A C and DB do not. Prove that the roads which intersect are together longer than those which do not.

16. The line that joins the vertex to the middle point of the base of a triangle is less than half the sum of the two sides.

17. The perimeter of a triangle is greater than the sum of the three middle lines-that is to say, the three lines joining the middle points of the sides with the opposite angles.

18. The perimeter of a triangle is less than double the sum of the three middle lines.

19. The perimeter of a triangle is greater than the sum of the straight lines which join any point in the interior of the triangle with the three angles, and is less than twice this sum.

20. The sum of the distances of a point in a straight line from two given points on the same side of the line is least when these distances make equal angles with the line.

21. A and B, on opposite sides of a line E F, are joined with a point C in E F. Prove that the difference between AC and B C is greatest when the angles ACE, BCF are equal.

CHAPTER V.

PARALLELS.

Definition.-Hitherto we have discussed the properties of those lines only which intersect. Straight lines may, however, be drawn in the same direction, in which case they will not intersect. Straight lines which lie in the same plane, but do not meet, however far they may be continued, are termed parallel lines.

This definition does not directly point out any method of drawing parallels. It may naturally be asked, "How do we

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T

Ꭰ

B

FIG. 59.

know that two straight lines drawn upon paper would never meet if they were continued far enough ?" In order to supply an answer to this question we must derive from the definition other properties of parallels.

Properties of Parallels.-Some of these properties will be best illustrated by descriptions of the usual methods of drawing parallels. Illustration 1.-Let a straight line, AB, be drawn on a sheet of paper, and then, by placing a ruler along A B and using a set-square, let two straight lines, CD, EF, be drawn, each perpendicular to A B. Now it is impossible that these two lines, CD and EF, which are perpendicular to the same straight line, shall meet; consequently they must be parallel.

E

B

FIG. 60.

The plan of erecting perpendiculars to the same straight line is one of the easiest ways of drawing parallels; but the method illustrates a particular case only of a more general proposition. Provided the two lines make the same angle with the given straight line, whether this angle be a right angle or not, the two straight lines will be parallel.

Illustration 2.-Take a set-square, having an angle V of any size, and, for convenience of reference, call the sides of the instrument which form the angle a and b, a being the longer side. Draw on paper a straight line, EF, and place the longer side, a, of the set-square in contact with the straight line E F, and along the other side, b, draw the line

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A B. Now remove the set-square to the other side of EF, placing the line a again in contact with E F, as in the figure, and draw the line CD.

B

FIG. 61.

α

Then AB and CD will be parallel. The two angles occupied by the square, being on opposite sides of EF, are termed alternate angles, and it will be proved that if we make a pair of alternate angles equal their sides will be parallel. Illustration 3.-Take a T-square with a double stock (Fig. 26), one part being

fixed at right angles to the arm and the other being movable about a screw. Let a sheet of paper be mounted on a drawing-board, and place the movable stock against the edge, A B, of the board

A

B

E

FIG. 62.

(Fig. 62), turning the arm of the T-square so as to make an acute angle with the stock. By drawing a pencil along the edge of the arm, make a straight line, CD, on the paper, and then, while keeping the angle between the arm and stock the same, slide the instrument to a new position, and draw a second straight line, EF. By this means we shall have two straight

lines, DC and FE, making equal angles with the straight line AB, and it will be found that these straight lines are parallel. Illustration 4.—In further illustration, let us again take the set-square,

D

FIG. 63.

F

and use one of its acute angles. Let A B be a straight line, and let it be required to draw a parallel to AB through the point O. Place the set-square so that one of its sides coincides with AB, and place a flat ruler, CD, in contact with another side. Keep

the ruler in the same position, and slide the set-square along it until the side passes through the point O, and, finally, draw the line OF. The lines BA and FO make, with CD, the angles BAD, FOD,

which are equal to one another, and in the theorems of this chapter it will be shown that when the corresponding angles B A D and FOD are equal, A B and E F are parallel.

Illustration 5.-Another method of drawing parallels depends upon the fact that two parallel straight lines are everywhere at the same distance. The principle is thus applied by carpenters :-Having well dressed one of the edges of a board, in order to cut the other parallel to it, the joiner measures upon a rod the required breadth of the board. Then, holding the rod at this distance from its end to the edge of the board with one hand, and, with the other, pressing a pencil against the end of the rod, he slides it along, and so marks upon the board a line parallel to the prepared side. Greater accuracy is obtained by the use of a special instrument called the carpenter's gauge (Fig. 64). In prin. ciple it is nothing more than the rod above mentioned, more strictly guided, and armed with a marking-point. Fig. 64 shows its shape. A smooth square block of hard wood, termed the guide, G, is placed upon a square bar, T, upon which it slides easily; at one end of the bar is a sharp point. By driving in the key, C, the bar and guide are locked together.

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When the piece of wood has been properly planed, a point in the required parallel is marked. Then the instrument is set so that, with the guide pressed against the edge of the board, the tracing point touches the mark that has been made, and the parallel is drawn by sliding the gauge along the edge of the board, with the guide pressed against one face and the point against the other. In the same manner any number of parallels may be drawn.

T

FIG. 64.

Denomination of Angles connected with Parallels.-When two parallel lines are cut by a third line eight angles are formed, which are named in pairs.

A pair of angles which have the same relative positions at the two points of intersection are termed corresponding angles (Fig. 65).

Two angles, one of which is vertically opposite to the angle corre sponding to the other, are termed alternate angles (Fig. 66 (a) (b). Interior angles on the same side are shown in Fig. 66 (c).

Propositions respecting Parallels.-The following propositions respecting parallels have to be proved :

(a). "When a straight line, falling across two other straight lines, makes two alternate angles equal, the two straight lines are parallel.” (b). "When a straight line, falling across two other straight lines, makes two corresponding angles equal, the two straight lines are parallel."

(c). "When a straight line, falling across two other straight lines, makes the two interior angles on the same side together equal to two right angles, the two straight lines are parallel."

And conversely

(a). "When a straight line falls across two parallel straight lines it makes any pair of alternate angles equal to one another."

(b). "When a straight line falls across two parallel straight lines it makes any pair of corresponding angles equal to one another."

(c). "When a straight line falls across two parallel straight lines it makes two interior angles on the same side equal to two right angles."