side AC, and the bisector of the angle BAC meet BC at D: shew that BD is less than CD.

39. If ABC is a triangle having the side AB less than the side

AC: shew that the bisector of the angle BAC lies between AB and the straight line drawn from A to the middle point of BC.

*40. Two points, A and B, lie on the same side of the straight line CD; P is a point in CD, such that AP and BP make equal angles with CD; Q is any other point in CD: shew that the sum of AP and BP is less than the sum of AQ and BQ.

SECTION III

PARALLELS AND PARALLELOGRAMS

DEF. 33. Parallel straight lines are such as are in the same plane and being produced to any length both ways do not meet. DEF. 34. When a straight line intersects two other straight lines it makes with them eight angles, which have received special names in relation to the lines or to one another.

Thus in the figure 1, 2, 7, 8 are called exterior angles, and 3, 4, 5, 6 interior angles; again 4 and 6, 3

and 5, are called alternate angles; lastly, 1 and 5, 2 and 6, 3 and 7, 4 and 8, are called corresponding angles.

THEOR. 21. If a straight line intercepts two other straight lines and makes the alternate angles equal, the straight lines are parallel.

Let the straight line EH intersect the straight lines AB, CD at F and G so as to make the angle AFG equal to the alternate angle FGD:

then shall AB and CD be parallel.

For if AB and CD are not parallel,

they will meet if produced far enough either towards B and D or towards A and C.

Suppose them to meet at a point K ;

then, of the two angles AFG and FGD, the one is an exterior,

and the other an interior opposite angle, of the triangle FGK; therefore the angle AFG is not equal to the angle FGD,

I. 9.

but by hypothesis the angle AFG is equal to the angle FGD, hence the angles AFG and FGD are both equal and unequal, which is absurd;

therefore AB and CD cannot meet,

that is, they are parallel.

Q.E.D.

Ex. 41. Show that the contrapositive form of Theor. 21 is included in Theor. 9.

THEOR. 22. If a straight line intersects two other straight lines and makes either a pair of alternate angles equal, or a pair of corresponding angles equal, or a pair of interior angles on the same side supplementary; then, in each case, the two pairs of alternate angles are equal, and the four pairs of corresponding angles are equal, and the two pairs of interior angles on the same side are supplementary.

Let the straight line EFGH intersect the straight lines AB, CD and make the alternate angles AFG, FGD equal:

A

then shall the alternate angles BFG, FGC be equal, the corresponding angles EFB, FGD equal, and the interior angles BFG, FGD supplementary.

I. 2.

The angle BFG is the supplement of the angle AFG, the angle FGC is the supplement of the angle FGD, and the angle AFG is equal to the angle FGD,

I. 2.

Hyp.

therefore the angle BFG is equal to the angle FGC. I. 1. Cor. 3. Again, the angle EFB is equal to the vertically opposite angle AFG,

I. 4.

and the angle FGD is also equal to the angle AFG, therefore the angle EFB is equal to the angle FGD.

Thirdly, the angle BFG is the supplement

of the angle AFG,

and the angle AFG is equal to the angle FGD,

Hyp. Ax. c.

1. 2.

Hyp.

therefore the angle BFG is the supplement of the angle FGD.

In like manner the other parts of the Theorem may be demonstrated.

Q.E.D.

COR. If a straight line intersects two other straight lines and makes a pair of corresponding angles equal, or a pair of interior angles on the same side supplementary, the straight lines are parallel.

By I. 21.

*Ex. 42. Straight lines that are perpendicular to the same straight line are parallel to one another.

AXIOM 3. Through the same point there cannot be more than one straight line parallel to a given straight line.

THEOR. 23. If two straight lines are parallel, and are intersected by a third straight line, the alternate angles are equal. Let the parallel straight lines AB, CD be intersected by the straight line EFGH:

then shall the angle AFG be equal to the alternate angle FGD, and the angle BFG equal to the alternate angle FGC.

Let A'B' be a straight line through F such that the angle A'FG is equal to the alternate angle FGD,

then A'B' is parallel to CD,

but AB is also parallel to CD,

and there is but one parallel to CD through the point F, therefore A'B' must fall along AB,

I. 21.

Hyp.

Ax. 3.

and therefore the angles AFG and A'FG are the same angle, therefore the angle AFG is equal to the alternate angle FGD. In like manner it may be shown that the angle BFG is equal

to the alternate angle FGC.

Q.E.D.

COR. 1. If a straight line intersects two parallel straight lines, and is perpendicular to one of them, it is also perpendicular to the other.

COR. 2. If a straight line intersects two parallel straight lines, it makes the corresponding angles equal, and the interior angles on the same side supplementary.

By I. 21.

COR. 3. If a straight line falling on two other straight lines makes the interior angles on the same side together less than two right angles, the two straight lines will meet, if continually produced, on the side on which are the angles which are less than two right angles.

*Ex. 43. If two straight lines are respectively parallel to two other straight lines, the angles made by the first pair are respectively equal to the angles made by the second pair.