115. If two lines are cut by a transversal, making the alternate interior angles equal, the lines are parallel.

Let AB and CD be cut by the transversal EF, making Z1=22.

To Prove AB and CD parallel.

Proof. From M, the middle point of so, draw MH to CD, and prolong MH until it meets AB in some point G.

Prove the AGMO and MSH equal in all respects.

116. COROLLARY.

If two lines are cut by a transversal, making any one of the following six cases true, the lines are parallel.

1. The alternate interior angles equal.

2. The alternate exterior angles equal.

3. The corresponding angles equal.

4. The sum of the interior angles on the same side equal to two R.A.'s.

5. The sum of the exterior angles on the same side equal to two R.A.'s.

6. The sum of the alternate interior and exterior angles equal to two R.A.'s.

Work this exercise by making the alternate exterior angles equal; also by making the corresponding angles equal.]

119. EXERCISE. The sum of two angles of a triangle cannot equal two right angles.

120. EXERCISE. The bisectors of the equal angles 1 and 2 in the figure of § 118, are parallel.

PROPOSITION XIX. THEOREM

121. If two parallels are cut by a transversal, the alternate interior angles are equal.

E

Let the parallel lines AB and CD be cut by the transversal EF.

To Prove AOS = OSD.

Proof. Suppose AOS is not equal to Osd.

Draw GH through 0, making ≤ GOS = ≤ OSD.

GH and CD are parallel. (?)

AB and CD are parallel. (?)

Through there are two parallels to CD, which contradicts (?).

.. The supposition that AOS and OSD are unequal, etc.

Q.E.D.

122. COROLLARY I. If two parallels are cut by a transversal, the six cases of § 116 are true.

123. COROLLARY II. If a line is perpendicular to one of two parallels, it is perpendicular to the other also.

124. EXERCISE. The bisectors of two alternate exterior angles, formed by a transversal cutting two parallel lines, are parallel.

125 EXERCISE. If a line joining two parallels is bisected, any other line through the point of bisection, and joining the parallels, is also bisected.

126. EXERCISE. If AB and CD are parallel (§ 117), and n = =1% R.A., find the values of the other seven angles.

PROPOSITION XX. THEOREM

127. If two lines are cut by a transversal, making the sum of the interior angles on the same side less than two right angles, the lines will meet if sufficiently produced.

E

A

Let AB and CD be cut by EF, making ≤1+≤ 2 < 2 R.A.'s. To Prove that AB and CD will meet.

Proof. If AB and CD do not meet, they are parallel. (?)

If they are parallel, ≤1+22=2 R.A.'s. (?)

This contradicts (?).

.. they cannot be parallel and must meet.

Q.E.B

128. COROLLARY. If two lines are cut by a transversal, making any one of the six cases of § 116 untrue, the lines will meet if sufficiently produced.

129. EXERCISE. The bisectors of any two exterior angles of a triangle will meet.

130. DEFINITION. Each angle, viewed from its vertex, has a right side and a left side.

AB is the right side of ABC, and BC is its left side.

B

131. If two angles have their sides parallel, right side to right side, and left side to left side, the angles are equal.

Let 1 and 2 have their sides parallel, right side to right side, and left side to left side.

Proof. Prolong AB and EF until they intersect.