SECTION VII.

PARALLEL STRAIGHT LINES.

THE SUM OF THE ANGLES OF A TRIANGLE ETC.

107. DEF. Straight lines in the same plane, which are such that they do not meet however far they are produced either way, are said to be parallel to each other.

108. When a straight line DAB crosses another straight line FAG a figure is formed in which there are four angles.

/D

G

B

Any one of these angles (DAG) is equal to its vertically opposite angle (FAB) and is the supplement of each of its adjacent angles (FAD, GAB).

When a straight line DABE crosses two other straight lines FAG, HBK we have two sets of four angles each, one set at A and another set at B.

We are about to prove that when the angles at A are equal respectively to the angles at B, then the lines FG, HK are parallel.

Also conversely when the lines FG, HK are parallel the angles at A and B are equal respectively.

That is to say, if the part of the figure consisting of the lines FAG and AD, were shifted so that AD moved along BA until A coincided with B, then, when FG and HK are parallel, FG will coincide with HK; and not otherwise.

109. Now, whether FG and HK are parallel or not, the angles at A and B have certain names indicating their relative positions.

The four angles which AB makes with the lines FG, HK are called interior angles.

The four angles which the lines AD, BE make with FG, HK are called exterior angles.

The interior angles are arranged in pairs called pairs of alternate angles.

Thus, FAB, ABK are alternate angles,

and GAB, ABH are alternate angles.

With reference to each exterior angle two of the interior angles are called adjacent and opposite respectively.

Thus, with reference to the exterior angle DAG

the interior adjacent angle is GAB

the interior opposite angle is ABK.

Also each of the four angles at B is said to correspond

to one of the angles at A.

Thus, DAG, ABK are corresponding angles

FAB, HBE are corresponding angles.

Example. If a straight line DABE cross two other straight lines FAG, HBK which are not parallel, the alternate angles are not equal.

E

K

Let C be the point in which the lines FAG, HBK meet.

Then ACB is a triangle ;

And of the alternate angles FAB, ABK, one is the exterior and the other an interior opposite angle of the triangle ABC.

And therefore one of the angles FAB, ABK is greater than the other.

Q.E.D.

[If the lines meet towards G and K, FAB is the exterior angle. If the lines meet towards F and H, ABK is the exterior angle. Also since if the lines meet towards GK the angle FAB is greater than ABK, it follows that GAB, ABK are in this case together less than two right angles.]

EXAMPLES XXXI.

1. Prove that when a straight line crosses two lines which are not parallel the two interior angles on that side of the line on which the two lines meet together are less than two right angles.

2. Prove that when a straight line crosses two other lines which are not parallel, the two interior angles on the opposite side of the line to the one which the two lines meet, together are greater than two right angles.

3. Prove that when a straight line crosses two lines which are not parallel the exterior angle on that side of the line on which the two lines meet is greater than the interior opposite angle.

4. In the figure of Question 3 prove that the exterior angle on the other side of the line is less than the interior opposite angle.

5. Two straight lines CAGF, CBKH which intersect at C, cross two parallel lines DABE, LGKM as in the figure.

Name the angles interior with respect to the lines CF, CH opposite (i) to DAC (ii) to CBE (iii) to BKM (iv) to HKM (v) to LGF.

6. In the above figure name the angles which are exterior to the lines CF, CH; also name the angles which are exterior to the lines DE, LM.

7. The angles (i) DAF, (ii) GKH, in the above figure, are each exterior to one pair of lines and exterior to another pair which are the pairs of lines in each case?

8. Prove that in the above figure the two interior angles FGK, GHK are together greater than two right angles.

9. In the above figure prove that the exterior angle LGF is less than the interior opposite angle GKH.

10. Name two pairs of interior angles which in the above figure are together less than two right angles and two other pairs which are together greater than two right angles.

11. The two equal sides CA, CB and the base BC of an isosceles triangle are produced, forming a figure in which the two lines CAD, CBE are crossed by a third line FBCG; prove that the exterior angles on the same side of BC are equal, and that the interior angles on the same side of BC are equal.

Proposition 27.

110. When a straight line, which crosses two other straight lines, makes the alternate angles equal, then the two straight lines are parallel.

Let the straight line DABE cross the two lines FAG, HBK ; and let the alternate angles FAB, ABK be equal;

it is required to prove

that the two straight lines FAG, HBK are parallel.

Now FG and HK are either parallel or they are not; if they are not parallel they meet, when produced, either towards G and K, or towards F and H. Now, because the alternate angles FAB, ABK are equal, therefore FG and HK do not meet at a point C, towards G, K; for, if they did,

then ABC would be a triangle whose side CA is produced, so that FAB an exterior angle of the triangle ABC would be greater than the interior opposite angle ABK. [Prop. 16.] Similarly, because the angles FAB, ABK are equal, therefore FG and HK do not meet towards F and H.

Therefore FG and HK are parallel.

Wherefore, when a straight line, etc.

Q.E.D.