Congruent Angles Prove Parallelism

In this section, we will prove that if two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel.

Instead of using a direct proof, we shall use proof by contradiction. Assume that the two lines, say $m$ and $n$, instersect at a point $C$ and are thus not parallel.

Note that $\angle{CBA}$ is supplementary to $\angle{CBD}$. We know that $\angle{CBD} = \angle{CAB}$, so let's call their measure $x$. Then we have $\angle{CBA} = 180^{\circ} - x$.

But notice something about triangle $ABC$! What do we know about the angles in a triangle?

The sum of the angles in a triangle is $180^{\circ}$! So we know that $$\begin{align*} 180^{\circ} &= \angle{CBA} + \angle{CAB} + \angle{ACB} \\ 180^{\circ} &= (180^{\circ} - x) + x + \angle{ACB} \\ 0^{\circ} &= \angle{ACB} \\ \end{align*}$$

But the only way for $\angle{ACB} = 0^{\circ}$ is if the two lines coincide with each other! Instead, they intersect each other at a point! Therefore, the two lines could not intersect; they must be parallel. $\Box$

Note that they can't coincide if $A$ and $B$ are distinct points.

Two of the Five ways to Prove that Two Lines are Parallel are:

1. Show that a pair of corresponding angles are congruent

2. Show that a pair of alternate interior angles are congruent

3. Show that a pair of same-side interior angles are supplementary

4. In a plane show that both alines are perpendicular to a third line

5. Show that both lines are parallel to a third line

For now we will be focusing on $1$ and $2$

To prove two lines are parallel by showing that a pair of corresponding angles are congruent, we use the theorem.

Theorem: If two parallel lines are cut by a transversal, the corresponding angles are congruent.

Consider the figure below:

Given $\angle\overline{{LM}}$ || $\angle\overline{{CD}}$ (line $LM$ is parallel to line $CD$), $\angle{1}$ $\cong{\angle{2}}$. $\angle{FED}$ is congruent to $\angle{MBE}$, therefore lines LM and CD are parallel.

In this lesson we were trying to make congruent angles so that lines are parallel. And in this worksheet we will do just that! Have fun!