TRANSVERSALS.

Definition. A straight line intersecting two or more straight lines is called a transversal.

In the figure EF is a transversal of the two non-parallel lines AB and CD.

The angles AHI, BHI, CIH, and DIH are called interior angles, and the angles AHE, EHB, CIF, and FID are called exterior angles.

The angles AHI and HID, or BHI and HIC are called alternate-interior angles.

The angles AHE and DIF, or BHE and CIF are called alternate-exterior angles.

The angles AHE and CIH, AHI and CIF, EHB and HID, or BHI and DIF are called exterior-interior angles.

I.

PROPOSITION VIII.

Theorem. If two parallel lines are cut by a third straight line, the alternate-interior angles are equal; and conversely.

Exercises. I. Find the value of an angle (1) if it is double its complement; (2) if it is one-fourth of its complement.

II.

III.

Given (1) that A is double its complement.

2.

A = 2(rt.

3.

IV. Solution.

- A). By hypothesis.

../A 2 rt. /'s-2/A. By Distribu-
A=
tive Law of Multiplication.

4. Adding 2 A to these two equals, we have
3 A2 rt. /'s. By Ax. 2.
Art. . By Ax. 7.

5.

Let the student give the solution of (2).

Q. E. F.

2. Find the value of an angle (1) if it is three times its supplement; (2) if it is one-third of its supplement.

3. How many degrees in the angle formed by the hands of a clock at 2 o'clock? 3 o'clock? 4 o'clock? 9 o'clock?

PROPOSITION IX.

I. Theorem. If two parallel lines are cut by a third straight line, the exterior angles are equal, and con

versely.

Let the student give the demonstration and state and prove the corollaries, if any, coming under the theorem.