Math Core

In the last lesson you started with parallel lines and concluded things about angles. Carpenters, surveyors and engineers usually need the opposite: they can measure angles, and they want to know whether two lines are parallel. That calls for the converses of the theorems you just learned.

Turning the theorems around

Recall that the converse of "if $p$ , then $q$ " is "if $q$ , then $p$ ." A true statement can have a false converse, so each converse needs its own justification. Just as before, one statement is accepted as a postulate and the rest are proved from it.

Ways to prove two lines parallel

If two lines are cut by a transversal and any one of the following is true, then the lines are parallel.

Corresponding Angles Converse (postulate): a pair of corresponding angles is congruent.
Alternate Interior Angles Converse: a pair of alternate interior angles is congruent.
Alternate Exterior Angles Converse: a pair of alternate exterior angles is congruent.
Consecutive Interior Angles Converse: a pair of consecutive interior angles is supplementary.

Here is how the Alternate Interior Angles Converse follows from the postulate, using the diagram below.

Lines m and n cut by transversal t. Nothing is assumed about whether m and n are parallel.

Given: $\angle 3 \cong \angle 6$ . Prove: $m \parallel n$ .

statement	reason
1. $\angle 3 \cong \angle 6$	Given
2. $\angle 2 \cong \angle 3$	Vertical Angles Congruence Theorem
3. $\angle 2 \cong \angle 6$	Transitive Property of Congruence
4. $m \parallel n$	Corresponding Angles Converse

Notice the structure: use known facts to turn the given pair into a corresponding pair, then apply the postulate.

Two more tools

Angle pairs aren't the only way to show lines are parallel. Two theorems let you reason from other parallel or perpendicular lines.

Parallel and perpendicular theorems

Transitive Property of Parallel Lines: if two lines are each parallel to a third line, then they are parallel to each other. If $p \parallel q$ and $q \parallel r$ , then $p \parallel r$ .
Lines Perpendicular to a Transversal Theorem: in a plane, if two lines are both perpendicular to the same line, then they are parallel to each other.

The second theorem is a special case of the Corresponding Angles Converse: if $a \perp t$ and $b \perp t$ , then the corresponding angles are both $90^\circ$ , so they are congruent. The words "in a plane" matter. In three dimensions, two lines perpendicular to the same line can be skew; think of two edges of a box that meet the same vertical edge, one at its top and one at its bottom, and point in different directions.

Choosing the right transversal

When a figure has more than two lines, every angle pair belongs to exactly one transversal, and the conclusion is about the other two lines. To identify them, look at the sides of the two angles. The line that forms a side of both angles is the transversal. The remaining sides lie on the two lines you can prove parallel.

In the diagram below, lines $p$ and $q$ are cut by two other lines, $r$ and $s$ . Four angles are numbered.

Four lines: p and q (horizontal), r and s (slanted).

$\angle 1$ and $\angle 2$ both have a side on $r$ . The transversal is $r$ , and the pair is corresponding, so $\angle 1 \cong \angle 2$ would prove $p \parallel q$ .
$\angle 1$ and $\angle 3$ both have a side on $p$ . The transversal is $p$ , the pair is corresponding, and $\angle 1 \cong \angle 3$ would prove $r \parallel s$ .

Worked example: Finding the value that makes lines parallel

In the first diagram, $m\angle 3 = (2x + 8)^\circ$ and $m\angle 6 = (4x - 40)^\circ$ . For what value of $x$ is $m \parallel n$ ?

$\angle 3$ and $\angle 6$ are alternate interior angles. By the Alternate Interior Angles Converse, the lines are parallel when these angles are congruent:

\begin{aligned} 2x + 8 &= 4x - 40 \\ 48 &= 2x \\ x &= 24 \end{aligned}

When $x = 24$ , both angles measure $56^\circ$ , so $m \parallel n$ .

Worked example: Supplementary angles as evidence

In the first diagram, $m\angle 4 = 118^\circ$ and $m\angle 6 = 64^\circ$ . Is $m \parallel n$ ?

$\angle 4$ and $\angle 6$ are consecutive interior angles. The lines would be parallel only if these angles were supplementary. But $118 + 64 = 182 \ne 180$ , so $m$ and $n$ are not parallel. (In fact, the Consecutive Interior Angles Theorem from the last lesson says that if they were parallel, the sum would have to be $180^\circ$ .)

Worked example: Which lines are parallel?

In the four-line diagram, you are told $\angle 2 \cong \angle 4$ . Which lines, if any, must be parallel?

$\angle 2$ has sides on $q$ and $r$ . $\angle 4$ has sides on $q$ and $s$ . The shared line is $q$ , so $q$ is the transversal and the conclusion is about $r$ and $s$ .

Both angles lie between $r$ and $s$ (right of $r$ , left of $s$ ), and one is above $q$ while the other is below it. They are alternate interior angles. By the Alternate Interior Angles Converse, $r \parallel s$ . Nothing can be concluded about $p$ and $q$ .

Worked example: A two-column proof

Use the first diagram. Given: $\angle 1$ and $\angle 7$ are supplementary. Prove: $m \parallel n$ .

statement	reason
1. $\angle 1$ and $\angle 7$ are supplementary	Given
2. $\angle 5$ and $\angle 7$ are supplementary	Linear Pair Postulate
3. $\angle 1 \cong \angle 5$	Congruent Supplements Theorem
4. $m \parallel n$	Corresponding Angles Converse

Step 3 works because $\angle 1$ and $\angle 5$ are supplementary to the same angle, $\angle 7$ . This proof shows that congruent alternate exterior angles aren't the only exterior test: supplementary same-side exterior angles also prove lines parallel.

Common mistake

When you are trying to prove lines parallel, you may not use the theorems that start with "if two parallel lines are cut by a transversal." Those theorems assume what you are trying to prove, which is circular reasoning. Cite the converses instead. Also double-check the transversal: congruent angles prove that the two lines that are not the transversal are parallel.

Tip

A quick way to write reasons correctly: if the statement you are justifying ends in " $\parallel$ ," the reason should be a converse (or one of the two parallel/perpendicular theorems). If the statement is about angles and you already know the lines are parallel, the reason is an original theorem.

Practice

Problems that mention $m$ , $n$ and $t$ use the first diagram. Problems that mention $p$ , $q$ , $r$ and $s$ use the four-line diagram.

Practice 1

If $\angle 2 \cong \angle 7$ , which reason justifies the conclusion $m \parallel n$ ?

Practice 2

If $m\angle 3 = (3x + 14)^\circ$ and $m\angle 7 = (5x - 12)^\circ$ , what value of $x$ makes $m \parallel n$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

If $m\angle 4 = (6x + 2)^\circ$ and $m\angle 6 = (2x + 10)^\circ$ , what value of $x$ makes $m \parallel n$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Suppose $m\angle 3 = 62^\circ$ and $m\angle 5 = 118^\circ$ . Can you conclude that $m \parallel n$ ?

Practice 5

In the four-line diagram, suppose $\angle 3 \cong \angle 4$ . Which conclusion is justified?

Practice 6

Lines $a$ , $b$ , $c$ and $d$ all lie in one plane. You know $a \perp c$ , $b \perp c$ and $d \parallel a$ . Which statement must be true?

Practice 7

Complete the proof. Given: $\angle 1 \cong \angle 8$ . Prove: $m \parallel n$ .

statement	reason
1. $\angle 1 \cong \angle 8$	Given
2. $\angle 8 \cong \angle 5$	?
3. $\angle 1 \cong \angle 5$	Transitive Property of Congruence
4. $m \parallel n$	Corresponding Angles Converse

Which reason belongs in step 2?

Practice 8

In the four-line diagram, $m\angle 1 = (5y + 5)^\circ$ and $m\angle 3 = (8y - 40)^\circ$ . What value of $y$ guarantees that $r \parallel s$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.