Prove Lines Parallel with Transversals | Geometry | LevelKite

You already know how to start with parallel lines and find angle measures. This lesson runs the other way: you start with facts about angles and prove that the lines are parallel.

Parallel lines are lines in the same plane that never meet. You cannot confirm "never meet" by looking, because lines go on forever. A transversal gives you another route. It is a line that crosses two other lines in the same plane at two different points, and the angles it forms can settle the question.

The tool is a converse. The converse of "if P, then Q" is "if Q, then P." A converse is not automatically true, so each one you use must be a postulate or theorem of this course. A postulate is a statement accepted without proof, and a theorem is a statement proved from earlier facts.

Line

GH

crosses line

AB

at

E

and line

CD

at

F

. The figure shows angle

GEB

and angle

EFD

and gives no size for either.

Line $GH$ is the transversal. No arrows appear on lines $AB$ and $CD$ , so nothing says they are parallel, even if they look that way. The figure is not drawn to scale, and you may use only what is marked or stated.

The figure shows $\angle GEB$ and $\angle EFD$ , a pair of corresponding angles: each holds the same position at its own intersection. Suppose you were told these two angles are congruent, meaning they have equal measures. Then the two lines would leave the transversal at the same tilt and head the same way, so they would never meet. The Converse of the Corresponding Angles Postulate accepts this idea as a starting point.

We write $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ for "line $AB$ is parallel to line $CD$ ," and $\cong$ for "is congruent to."

Key concept

Angle fact first, parallel lines second

Suppose a transversal crosses two lines in a plane. The lines are parallel when one of these conditions holds: a pair of corresponding, alternate interior or alternate exterior angles is congruent, or a pair of consecutive interior angles is supplementary, meaning their measures add to $180^\circ$ . This is the converse direction, shown below for the corresponding pair in the figure. The theorem direction goes from parallel lines to angle facts, and it applies only when the lines are given parallel.

\angle GEB \cong \angle EFD \;\Longrightarrow\; \overleftrightarrow{AB} \parallel \overleftrightarrow{CD}

Any one condition proves the two lines parallel. Each pair has one angle at each intersection.

Angle pair	Where the angles sit	Example in the figure	Condition needed	Reason to name
Corresponding	Same position at each intersection	$\angle GEB$ and $\angle EFD$	Congruent	Converse of the Corresponding Angles Postulate
Alternate interior	Between the two lines, on opposite sides of the transversal	$\angle AEF$ and $\angle EFD$	Congruent	Alternate Interior Angles Converse
Alternate exterior	Outside the two lines, on opposite sides of the transversal	$\angle GEA$ and $\angle DFH$	Congruent	Alternate Exterior Angles Converse
Consecutive interior	Between the two lines, on the same side of the transversal	$\angle BEF$ and $\angle EFD$	Supplementary	Consecutive Interior Angles Converse

Worked example

Decide which facts prove the lines parallel

Use the first figure, where line $GH$ crosses line $AB$ at $E$ and line $CD$ at $F$ . Taking each fact by itself, which ones prove $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ ? (a) $\angle GEB \cong \angle AEF$ ; (b) $\angle GEA \cong \angle DFH$ ; (c) $\angle AEF \cong \angle EFC$ ; (d) $\angle BEF$ and $\angle EFD$ are supplementary.

Fact (a). Both angles sit at E, and they are vertical angles, the opposite angles formed where two lines cross. The Vertical Angles Theorem makes them congruent whether or not the lines are parallel, and neither angle involves line CD. This fact proves nothing about the two lines.
$\angle GEB \cong \angle AEF$
Fact (b). Angle GEA is at E and angle DFH is at F. Both lie outside the two lines, on opposite sides of the transversal, so they are alternate exterior angles. They are congruent, which is their condition, so the Alternate Exterior Angles Converse proves the lines parallel.
$\angle GEA \cong \angle DFH$
Fact (c). Both angles lie between the two lines, on the same side of the transversal, so they are consecutive interior angles. Their condition is supplementary, not congruent. Two 80-degree angles are congruent but add to 160 degrees, so congruence alone does not meet the condition.
$\angle AEF \cong \angle EFC$
Fact (d). These are consecutive interior angles on the other side of the transversal. They are supplementary, which is their condition, so the Consecutive Interior Angles Converse proves the lines parallel.
$\angle BEF \text{ and } \angle EFD \text{ are supplementary}$

Answer

Facts (b) and (d) each prove $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ by a converse, while facts (a) and (c) do not.

Now the angle measures depend on a variable. We write $m\angle AEF$ for the measure of $\angle AEF$ . The value that makes two lines parallel is the value that meets the angle condition, because that condition is what lets you use a converse.

Choose your equation from the type of pair:

For a corresponding, alternate interior or alternate exterior pair, set the two measures equal.
For a consecutive interior pair, set the sum of the two measures equal to $180^\circ$ .

After you solve, substitute the value back and check that each measure is between $0^\circ$ and $180^\circ$ . The figure below has no arrows on lines $AB$ and $CD$ , because parallel is the result you want, not a given.

Angle

AEF

measures

(3x + 10)^\circ

and angle

EFD

measures

(5x - 30)^\circ

. These measures come from the text, not from the drawing.

Worked example

Find x for an alternate interior pair

In the figure, $m\angle AEF = (3x + 10)^\circ$ and $m\angle EFD = (5x - 30)^\circ$ . Find the value of $x$ that makes $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ .

Name the pair. Angles AEF and EFD lie between the two lines, on opposite sides of transversal GH, so they are alternate interior angles. The Alternate Interior Angles Converse needs them to be congruent, so set their measures equal.
$5x - 30 = 3x + 10$
Subtract 3x from both sides to gather the x terms on the left (Subtraction Property of Equality).
$2x - 30 = 10$
Add 30 to both sides to get the x term alone (Addition Property of Equality).
$2x = 40$
Divide both sides by 2 (Division Property of Equality).
$x = 20$
Check by substituting 20 into each expression. Both measures are 70 degrees, so the angles are congruent, and 70 is between 0 and 180.
$5(20) - 30 = 70 \quad\text{and}\quad 3(20) + 10 = 70$

Answer

When $x = 20$ , both angles measure $70^\circ$ , so $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ by the Alternate Interior Angles Converse.

Worked example

Find x for a consecutive interior pair

Use the same lines with new measures: $m\angle BEF = (3x - 10)^\circ$ and $m\angle EFD = (2x - 10)^\circ$ . Find the value of $x$ that makes $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ .

Angles BEF and EFD lie between the two lines, on the same side of the transversal, so they are consecutive interior angles. The Consecutive Interior Angles Converse needs them to be supplementary, so their measures must add to 180.
$(3x - 10) + (2x - 10) = 180$
Combine like terms on the left: 3x and 2x make 5x, and the two -10 terms make -20.
$5x - 20 = 180$
Add 20 to both sides (Addition Property of Equality).
$5x = 200$
Divide both sides by 5 (Division Property of Equality).
$x = 40$
Check by substituting 40. Both measures are between 0 and 180 degrees, and they add to 180.
$3(40) - 10 = 110,\quad 2(40) - 10 = 70,\quad 110 + 70 = 180$

Answer

When $x = 40$ , the angles measure $110^\circ$ and $70^\circ$ , so they are supplementary and $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ by the Consecutive Interior Angles Converse.

A proof is a chain of statements that leads from what you are given to what you must show. The Given is what you may start from, and the Prove is the statement you must reach. A two-column proof lists each statement beside its reason. Each reason is the exact name of a definition, postulate, property or previously proved theorem, or the word Given. How a figure looks is never a reason, and a step cannot use its own conclusion as its reason.

You have named the Alternate Interior Angles Converse as a reason. Now you will prove it from the postulate, so this proof cannot use it as a reason. Use the figure just before the value examples.

Given: $\angle AEF \cong \angle EFD$

Prove: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$

Plan: The Converse of the Corresponding Angles Postulate can finish the proof, so aim for a corresponding pair. You already saw that $\angle GEB$ and $\angle EFD$ are corresponding. Also, $\angle GEB$ and $\angle AEF$ are vertical angles, so $\angle GEB$ connects the Given to that pair.

Worked example

Walk through the proof

Line $GH$ crosses line $AB$ at $E$ and line $CD$ at $F$ . Given $\angle AEF \cong \angle EFD$ , prove $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ . Each step below pairs a statement with its reason.

Reason: Given. This is your starting fact. You do not yet know that the lines are parallel, so you cannot use a theorem that requires parallel lines.
$\angle AEF \cong \angle EFD$
Reason: Vertical Angles Theorem. These are the opposite angles formed where lines AB and GH cross at E. This theorem does not require parallel lines.
$\angle GEB \cong \angle AEF$
Reason: Transitive Property of Congruence. Angle GEB is congruent to angle AEF, and angle AEF is congruent to angle EFD, so angle GEB is congruent to angle EFD.
$\angle GEB \cong \angle EFD$
Reason: Converse of the Corresponding Angles Postulate. Angles GEB and EFD are corresponding angles for transversal GH, and they are congruent. This is the converse direction: the angle fact leads to parallel lines.
$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$

Answer

Angles $GEB$ and $EFD$ are congruent corresponding angles, so $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ . That proves the Alternate Interior Angles Converse, and you may name it as a reason from now on.

Two-column proof of the Alternate Interior Angles Converse

Statement	Reason
$\angle AEF \cong \angle EFD$	Given
$\angle GEB \cong \angle AEF$	Vertical Angles Theorem
$\angle GEB \cong \angle EFD$	Transitive Property of Congruence
$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$	Converse of the Corresponding Angles Postulate

The next proof uses both directions. In the figure below, the arrows show that lines $AB$ and $CD$ are given parallel. Lines $EF$ and $GH$ carry no arrows, because you must prove that they are parallel. Points $E$ and $G$ lie on line $AB$ , and points $F$ and $H$ lie on line $CD$ .

Given: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ and $\angle AEF \cong \angle GHD$

Prove: $\overleftrightarrow{EF} \parallel \overleftrightarrow{GH}$

Plan: Lines $AB$ and $CD$ are given parallel, so use the theorem direction with transversal $GH$ : alternate interior angles $\angle GHD$ and $\angle AGH$ are congruent. Connect this to the Given to get $\angle AEF \cong \angle AGH$ . Those are corresponding angles for lines $EF$ and $GH$ , with line $AB$ as the transversal, so finish in the converse direction.

The arrows state the Given that line

AB

is parallel to line

CD

. Angles

AEF

,

AGH

and

GHD

are shown, and only the Given says that angles

AEF

and

GHD

are congruent.

Two-column proof that line EF is parallel to line GH

Statement	Reason
$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$	Given
$\angle AEF \cong \angle GHD$	Given
$\angle GHD \cong \angle AGH$	Alternate Interior Angles Theorem
$\angle AEF \cong \angle AGH$	Transitive Property of Congruence
$\overleftrightarrow{EF} \parallel \overleftrightarrow{GH}$	Converse of the Corresponding Angles Postulate

Here is the central idea: to prove two lines parallel, start from a stated angle fact and name the matching converse. With a transversal crossing two lines in a plane, that fact is a congruent corresponding, alternate interior or alternate exterior pair, or a supplementary consecutive interior pair. To find the value that makes lines parallel, write the equation that condition requires, solve it and check the measures. Use a parallel-line theorem only when the lines are given parallel, and never decide that lines are parallel from how a drawing looks.