3. Show that the angles AOE and BOD are complementary.

4. Show that the angles AOD and COD are supplementary; and also that the angles BOE and COE are supplementary.

5. If the angle BOD is 37°, how many degrees are there in AOE?

6. If two angles are supplementary, and the greater is 9 times the less, how many degrees are there in each angle? 7. If an angle is 11 times its complement, how many degrees does it contain?

PARALLEL LINES.

68. Parallel straight lines are such as lie in the same plane, and never meet, however far they are produced in both directions.

69. A straight line crossing several other lines is called a transver

sal; as EF.

When two straight lines are cut by a transversal, eight angles are formed, which are named as follows:

The four angles a, b, g, h, without the two lines, are called exterior

angles.

E

The four angles c, d, e, f, within the two lines, are called interior angles.

The pair c and e, and the pair d and f, are called alternate-interior angles.

The pair a and g, and the pair b and h, are called alternate-exterior angles.

The pairs a and e, b and f, c and g, d and h, are called corresponding angles.*

*

* Called also exterior-interior angles.

Proposition 10. Theorem.

70. Two straight lines in the same plane, and perpendicular to a third straight line, are parallel to each other.

Proof. If AB and CD are not parallel, they will meet if sufficiently produced, and we shall have two perpendiculars from the same point to the same straight line, which is impossible.

(58)

Therefore they cannot meet.

... AB and CD are parallel.

(68)

Q.E. D.

EXERCISES.

1. Find the number of degrees in each of two angles if they are complementary, and the greater is three times the less.

2. Find the number of degrees in each of two angles if they are supplementary, and the greater exceeds the less by 40°.

3. Find the number of degrees in each of two angles if they are supplementary, and the less is one-fifth the greater.

Proposition 11. Theorem.

71. If a straight line is perpendicular to one of two parallels, it is also perpendicular to the other.

C

E

Hyp. Let AB and CD be two parallel lines, and let AC be to CD.

To prove

AC 1 to AB.

Proof. Through A where AC intersects AB, draw AE L to AC.

AE and CD are both 1 to AC,

Because

But

AE is parallel to CD.

AB is parallel to CD.

(70) (Hyp.)

... AE coincides with AB.

Through a given pt. only one line can be drawn || to a given line (Ax. 12). .. AB is to AC, and ... AC is to AB. Q. E. D.

EXERCISES.

1. Find the value of an angle if it is four times its complement.

2. Find the value of an angle if it is three times its supplement.

3. Find the value of an angle if it is one-eighth of its complement.

4. Find the number of degrees in each of two angles. if they are supplementary, and the greater is four times the less.

Proposition 12. Theorem.

72. If a straight line cut two parallel straight lines, the alternate-interior angles are equal.

Hyp. Let the straight line LM cut the || straight lines AB, CD, at the points E, F.

To prove

AEFEFD.

Proof. Through K, the middle point of EF, draw HG to AB.

HG is also to CD.

(71)

Turn the figure KEGD, in its own plane, about K as a pivot until KG coincides with its equal KH.

Then since

and

<FKG/EKH, being verticals, (55)

.. KF will coincide with KE,

and point F will coincide with point E.

(Cons.)

... GF to KG will coincide with HE to KH. (58)

..s KFG and KEH coincide;

../ KFC = /KEH.

Q.E.D.

73. COR. 1. The alternate-interior angles BEF, EFC

are also equal.

(50)

74. COR. 2. The alternate-exterior angles BEL, CFM are also equal.

(55) and (Ax. 1)

Proposition 13. Theorem.

75. Conversely, if a straight line cut two other straight lines, so as to make the alternate-interior angles equal, the two straight lines are parallel.

Hyp. Let the straight line EF cut the two straight lines AB, CD at G and H, so that AGH = /GHD.

Proof. Through H draw KL || to AB.

Then, since AB and KL are,

76. COR. If the alternate-exterior angles are equal, the

two lines are parallel.

If the sum of the two interior angles on the same side is less than two right angles, the lines will meet.

EXERCISE.

If a line is drawn through the vertex of an angle perpendicular to the bisector, prove (1) that it bisects the supplementary angle, and (2) that it makes equal angles with the sides of the given angle.