EXAMPLES XXXII.

1. If two straight lines are perpendicular to the same straight line they are parallel to each other.

2. In the figure of Prop. 16, prove that CF is parallel to AB.

3. If two finite straight lines bisect each other the lines joining their extremities in pairs are parallel.

4. Two lines drawn one from each of the two extremities of the base of a triangle and terminated by the opposite sides cannot bisect each other.

5. Two men are walking in Indian file along a straight path; one of them turns through a certain angle to his right and walks a certain distance straight, and then turns through an equal angle to his left and then walks straight on; shew that he can never meet the other man.

6. ABCD is a quadrilateral whose opposite sides are equal; prove that its opposite sides are parallel.

7. A quadrilateral has one pair of opposite sides equal, and one pair of opposite angles right angles; prove that its opposite sides are equal.

8. In the figure of Prop. 16, prove that AF is parallel to BC.

9. If in the figure of Prop. 16, AB is bisected in K and CK produced to L so that CK=KL, then LAF is a straight line.

Proposition 28.

111. When a straight line which crosses two other straight lines makes (i) an interior angle equal to an interior opposite angle, then the two straight lines are parallel; also when it makes (ii) the two interior angles on one side of it, together equal to two right angles, then the straight lines are parallel.

Let the line DABE crossing the two straight lines FAG, HBK, (I) make an exterior angle DAG equal to the interior opposite angle ABK,

or, (II) make GAB, ABK two interior angles on one side of DE together equal to two right angles;

it is required to prove

that, in either case, FG and HK are parallel.

(I) Since it is given that the angle DAG is equal to ABK and the angle FAB is equal to DAG, [Prop. 15.] therefore the angle FAB is equal to ABK;

and these are alternate angles,

therefore FG is parallel to HK.

[Prop. 27.]

(II) Since it is given that the angles GAB, ABK together are equal to two right angles,

therefore they are together equal to GAB, FAB; [Prop. 13.] taking away the common angle GAB, it follows that the angle FAB is equal to ABK ; and these are alternate angles,

therefore FG is parallel to HK.

Wherefore, when a straight line, etc.

[Prop. 27.]

Q.E.D.

NOTE. Prop. 28 is really only a Corollary to Proposi

tion 27.

The three tests of parallelism,

I. The alternate angles equal,

II. Exterior angle equal to interior opposite angle,

III. Two interior angles on the same side together equal to a straight angle,

are only different forms of the same thing.

Any two of them can be at once deduced from the third.

EXAMPLES XXXII

1. Assuming II. of the above tests deduce from it I. and III. 2. Assuming III. deduce from it I. and II.

3. Deduce II. directly from Prop. 16.

4. Deduce III. directly from Prop. 17.

5. D is a point in the side AB of the triangle ABC. The angle ADE is made equal to ABC on the same side of AB as ABC; prove that DE is parallel to BC.

6. When a straight line crossing two other straight lines makes the two interior angles on one side of the line together equal to the two interior angles on the other side of the line, the two straight lines are parallel.

7. When a straight line crossing two other straight lines makes the two exterior angles on the same side of the line together equal to two right angles the two lines are parallel.

8. When a straight line crossing two other straight lines makes the two exterior angles on one side of the line together equal to the two interior angles on the same side of the line the two straight lines are parallel.

112. Let two straight lines FG, HK be such that when DABE crosses them the angles GAB, ABK together make up two right angles.

In BK take a point C; join AC.

Between AG and AC draw the line AX.

By Prop. 28 we know that AG produced does not cut BK produced.

By Prop. 17, we can show that any line AC drawn from A to cut BK, makes the angle CAB less than GAB.

We have not proved, that there is no line AX, such that the angle XAB is less than GAB, which when produced will not cut BK produced.

This is the same thing as saying that we have not proved that there is no line AX, making the angle XAB less than GAB, which is parallel to BK.

This Euclid takes for granted.

We may consider this assumption either as a definition of non-parallelism or as an Axiom.

It is the celebrated 12th Axiom of Euclid.

It may be pointed out that the straight lines above referred to are unlimited. Finite straight lines which satisfy Axiom 12 do not necessarily meet.

LINES WHICH ARE NOT PARALLEL.

"Axiom 12."

107

113. When two straight lines, which are crossed by a third straight line, are such that the two interior angles on one side of the third line, are together less than two right angles, then the two straight lines will meet on that side of the third line.

F

K

H

'E

114. Thus, in the figure let the two straight lines GE, HK which are crossed by the third line DAB be such that the angles GAB, ABK together are less than two right angles.

Then Axiom 12 asserts not only that FG and HK are not parallel but also that they are such that they meet on the side of AB towards G and K.

Example. Assuming the following Axiom

'Through a given point one line and one line only can be drawn which is parallel to a given straight line,' prove Axiom 12.

Let A be a given point, HBK a given straight line.

Let the line FAG which passes through A, make the angles GAB, ABK together equal to two right angles. [See figure on p. 106.] Then, by Prop. 28, FAG is parallel to HBK.

Now, because AG is parallel to BK,

therefore (by our Axiom) no other line through A can be parallel to BK.

That is, any other line AX, through A will meet BK in some point. It is shewn in Example p. 100 that the two interior angles XAB, KBA on the same side of AB, are together less than two right angles on the side of AB on which the lines AX, BK meet; which proves Axiom 12.

Q.E.D.