Proving Lines are Parallel

It is amazing how you can use the measures of angles to determine if lines are parallel.

Picture of a bridge with many columns stretching over a forest

Vocabulary

  • transversal
  • interior angles
  • exterior angles
  • corresponding angles
  • alternate interior
  • alternate exterior
  • same side interior
  • same side exterior

In the last lesson we learned relationships about certain pairs of angles when two parallel lines are cut by a transversal.

This lesson is similar, except we will not be given that the two lines are parallel but instead we will be asked to prove if the two lines are parallel or not parallel based on information that is provided.

Alternate Exterior Angles Converse Theorem

First, let's look at information that will help us determine if the lines are parallel. We are going to use the converse of the theorems we learned from the lsat lesson.

Alternate Exterior Angles Converse Theorem states that if two lines are cut by a transversal and the alternate exterior angles are congruent, then the two lines are parallel.

Angle A and angle B are shown to be equal as alternate exterior angles

If ∠a ≅ ∠b then m||n.

Alternate Interior Angles Converse Theorem

Alternate Interior Angles Converse Theorem states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the two lines are parallel.

If ∠a ≅ ∠b then m||n.

Corresponding Angles Converse Theorem

Corresponding Angles Converse Theorem states that if two lines are cut by a transversal and the corresponding inter angles are congruent, then the two lines are parallel.

Angle A and angle B are shown to be equal and corresponding angles

If ∠a ≅ ∠b then m||n.

Converse Theorems

Remember that if the following angles are congruent, then the two lines are parallel:

Alternate interior angles

Two lines are cut by a transversal showing the alternate interior angles

Alternate exterior angles

Two lines are cut by a transversal showing the alternate exterior angles

Corresponding angles

Two lines are cut by a transversal showing the corresponding angles

Also, if the following angles are supplementary, then the two lines are parallel.

  • Same side interior
  • Same side exterior

These are all converse theorems from the last lesson. Have these written in your notes.

Same Side Interior Angles Converse Theorem

Same Side Interior Angles Converse Theorem states that if two lines are cut by a transversal and the same side interior angles are supplementary, then the two lines are parallel.

Two lines are cut by a transversal showing same side interior angles

If ∠a is supplemetary to ∠b then m||n

Same Side Exterior Angles Converse Theorem

Same Side Exterior Angles Converse Theorem states that if two lines are cut by a transversal and the same side exterior angles are supplementary, then the two lines are parallel.

Two lines are cut by a transversal showing same side exterior angles

If ∠a is supplemetary to ∠b then m||n

Example #1

Here is an example showing how to use this information. Are lines c and d parallel?

Line c and line d are a pair of lines that are cut by a transversal. The angle to the lower left of the intersection with d is 60 degrees. The angle to the lower left of the intersection with c is 60 degrees

Solution: First identify the relationship of the pair of angles that have given information. In the diagram, the angles that have a measure of 60° are corresponding angles. Are the lines parallel?

Answer: Yes, lines c and d are parallel because corresponding angles are congruent.

Answers

Yes or No will not be sufficient answers to these problems. Each answer has to be given a reason.

If the previous example had this information instead your answer would be: Lines m and n are not parallel because the corresponding angles are not congruent.

Line m and line n are a pair of lines that are cut by a transversal. The angle to the lower left of the intersection with m is 62 degrees. The angle to the lower left of the intersection with n is 60 degrees

Example #2

Are lines f and g parallel?

Line f and line g are a pair of lines that are cut by a transversal. The angle to the lower right of the intersection with g is 150 degrees. The angle to the upper right of the intersection with f is 30 degrees

Solution: Again, first identify the relationship of the angles that have given information. In the problem, angles 130° and 50° are same side exterior angles. Are the lines parallel?

Answer: Yes, lines f and g are parallel because same side exterior angles are supplementary.

Example #3

What would the value of x have to be for the lines f and g to be parallel?

Line g and line f are a pair of lines that are cut by a transversal. The angle to the lower right of the intersection with f is 140 degrees. The angle to the upper left of the intersection with g has value 7x

Solution: Again, first identify the relationship of the angles that have given information. In the problem, angles 140° and 7(x) are alterior exterior angles so they must be congruent for the lines to be parallel.

7x = ___

Answer: 140

x = ___

Answer: 20

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