Proposition 29.

115. When a straight line crosses two parallel straight lines, (i) it makes the alternate angles equal, (ii) it makes each exterior angle equal to the interior opposite angle on the same side of it, (iii) it makes the two interior angles on each side of it together equal to two right angles.

Let FG and HK be two parallel straight lines, and let DABE cross them,

it is required to prove

(I) that alternate angles FAB, ABK are equal;

(II) that an exterior angle DAG is equal to the interior opposite angle ABK ;

(III) that interior angles GAB, ABK, on one side of DE, together are equal to two right angles.

The angles GAB, ABK together must be either equal to, or less than, or greater than two right angles; Now, because FG and HK are parallel, therefore

GAB, ABK together are not less than two right angles, for then by Axiom 12, AG and BK would meet towards G, K. Also, because FG and HK are parallel, therefore

GAB, ABK together are not greater than two right angles, for if so, the angles FAB, ABH together would be less. than two right angles,

so that by Axiom 12, FG and HK would meet towards F, H;

therefore GAB, ABK together

are equal to two right angles.

(III.)

Again, because GAB, ABK together are equal to two right angles, therefore the angles GAB, ABK together

are equal to GAB, FAB;

[Prop. 13.]

taking away the common angle GAB, it follows that the angle FAB is equal to the alternate angle ABK. (I.) Again, since the angle DAG is equal to FAB, [Prop. 15.] therefore also the angle DAG is equal to

the interior opposite angle ABK.

Wherefore, when a straight line crosses, etc. Q.E.D.

EXAMPLES XXXIV.

(II.)

1. A line drawn parallel to the base of a triangle makes angles with the sides equal to the angles at the base.

2. Any two points A, B being taken in the parallel lines FAG, HBK, AB is joined and bisected in C, prove that any straight line LCM drawn through C and terminated by the parallels is bisected at C.

3. Prove that the line drawn through C parallel to the opposite side of the triangle ABC makes the three angles at C equal to the three angles of the triangle.

4. A quadrilateral whose opposite sides are parallel, has its opposite angles equal, and its adjacent angles supplementary.

5. If one pair of parallel straight lines cross another pair of parallel lines, then every angle between the lines is either equal or supplementary to any other.

6. If one of the angles of a quadrilateral whose opposite sides are parallel is bisected by a diagonal all the sides of the quadrilateral are equal.

7. From the same point only one line can be drawn parallel to a given line.

8. If from any point in the bisector of an angle a straight line be drawn parallel to either of the lines making the angle, the triangle thus formed will be isosceles.

Proposition 30.

116. Straight lines which are parallel to the same straight line are parallel to one another.

Let the straight lines EF, GH be each parallel to KL,

it is required to prove

that EF, GH are parallel to each other.

Let a line ABD cross the lines EF, GH, KL

cutting them in A, B, D respectively.
Then, because EAF, KDL are parallel, therefore
the alternate angles EAD, ADL are equal;

[Prop. 29.]
and because GBH, KDL are parallel, therefore
the exterior angle ADL is equal to

the interior opposite angle DBH; [Prop. 29.] wherefore the angle EAB is equal to the angle ABH ; hence because the alternate angles EAB, ABH are equal, therefore EF is parallel to GH. [Prop. 27.] Wherefore, straight lines which are parallel, etc. Q.E.D. EXAMPLES XXXV.

1. A series of parallel straight lines are drawn; prove that if a straight line cut any one of these parallels it must cut all of them.

2. A series of parallel straight lines is crossed by another series of parallel straight lines; prove that all the angles of the figure are either equal or supplementary.

3. If two straight lines which are parallel to the same straight line, each pass through the same point, they must be portions of the same straight line.

Proposition 32.

117. I. An exterior angle of a triangle is equal to the two interior opposite angles together.

II. The three interior angles of a triangle together are equal to two right angles.

Let CAD represent an exterior angle of a triangle ABC, it is required to prove

I. that the angle CAD is equal to ABC, BCA together; II. that the three angles CAB, ABC, BCA together make up two right angles.

E

Д

B

A

I. Let AE be the line through A parallel to BC.
Then, because BC, AE are parallel,

and the angles BCA, CAE are alternate,
therefore the angles BCA, CAE are equal; [Prop. 29.]
Also, because BC, AE are parallel,

therefore the exterior angle EAD

is equal to the interior opposite angle ABC. [Prop. 29.] Hence, the whole angle CAD

is equal to the angles ABC, CAB together.

II. Now, to each of these equals add the angle CAB, then the angles BAC, CAD together are equal to

the three angles CAB, ABC, BCA together;

and the angles BAC, CAD together are two right angles. Therefore also the three angles CAB, ABC, BCA together are two right angles.

Wherefore, an exterior angle of a triangle, etc. Q.E.D.

EXAMPLES XXXVI.

1. If one angle of a triangle is a right angle, the other two angles together make up a right angle.

2. If one angle of a triangle is equal to the sum of the other two, the triangle is a right-angled triangle.

3. The angle of an equilateral triangle contains 60 degrees. 4. The angles of an isosceles right-angled triangle contain 90 degrees, 45 degrees and 45 degrees respectively.

5. ABC is a right-angled triangle having a right angle at B; the angle ABD is equal to BAC; prove that the angle CBD is equal to BCA.

6. A point D is taken in the hypotenuse AC of a rightangled triangle ABC such that AD is equal to DB; prove that D is the middle point of AC.

7. A series of right-angled triangles have a common hypotenuse, shew that the vertices are all equidistant from the middle point of the common hypotenuse.

8. A triangle has two equal angles each of which is half the third angle; what is the size of the angles?

9. A triangle has two equal angles each of which is a quarter of the third angle; what is the size of the angles ?

10. The angles of a triangle are proportional to 1, 2, 3 respectively; how many degrees do they contain?

11. ABC is a triangle right-angled at A; D is the middle point of BC and AK is perpendicular to BC; prove that the angle DAK is equal to the difference between the angles at B, C.

12. An angle of a triangle is acute, right or obtuse according as it is less than, equal to or greater than the other two angles together.

13. D is the middle point of the side BC of a triangle ABC; prove that the angle BAC is acute, right or obtuse according as AD is greater than, equal to or less than half BC.