Topics
2. Parallel and Perpendicular Lines
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United States of America
Grades 9-12

2.02 Proving lines parallel

Lesson
Practice

Introduction

We learned about the angle relationships created when lines and parallel lines are cut by a transversal in  2.01 Parallel lines and transversals  . We will now use those relationships and their converses to prove that lines are parallel and construct parallel lines using various methods.

Proving lines parallel

To determine if two lines are parallel, we can use the converses of the postulate and theorems which relate angle pairs formed by two lines and a transversal.

Converse of corresponding angles postulate

If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

A pair of parallel lines intersected by a transversal. Two marked angles lie on the same side of the transversal, and on the same sides of the parallel lines. The two angles are congruent.
Converse of consecutive interior angles theorem

If two lines and a transversal form consecutive interior angles that are supplementary, then the lines are parallel.

A pair of parallel lines intersected by a transversal. Two marked angles lie on the same side of the transversal, between the parallel lines. The two angles are supplementary.
Converse of consecutive exterior angles theorem

If two lines and a transversal form consecutive exterior angles that are supplementary, then the lines are parallel.

A pair of parallel lines intersected by a transversal. Two marked angles lie on the same side of the transversal, outside the parallel lines. The two angles are supplementary.
Converse of alternate interior angles theorem

If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.

A pair of parallel lines intersected by a transversal. Two marked angles lie on the opposite sides of the transversal, between the parallel lines. The two angles are congruent.
Converse of alternate exterior angles theorem

If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel.

A pair of parallel lines intersected by a transversal. Two marked angles lie on the opposite sides of the transversal, outside the parallel lines. The two angles are congruent.

Using properties of parallel lines, we can relate three parallel lines given one line in common.

Transitivity of parallelism

If a\parallel b and b\parallel c, then a\parallel c.

Lines a, b, and c. Line a is parallel to line b. Line b is parallel to line c.

Examples

Example 1

Determine whether or not there is a pair of parallel lines in the figure.

Two lines cut by a transversal. The interior angles on the same side of the transversal have a measure of 124 degrees and 46 degrees.
Worked Solution
Create a strategy

We can see that the two marked angles form a pair of consecutive interior angles. This means that we can use the converse of consecutive interior angles theorem to check whether or not we have parallel lines.

Apply the idea

If we add the measures of the consecutive interior angles together we get:

124\degree+46\degree=170\degree

Since the sum of the measures is not 180 \degree, the two angles are not supplementary.

The converse of consecutive interior angles states that the lines are parallel if the consecutive interior angles are supplementary. Since they are not supplementary, the lines are not parallel.

Therefore, there is not a pair of parallel lines in the figure.

Example 2

Find the value of x required for the figure to contain a pair of parallel lines.

Two lines cut by a transversal. The exterior angle on the left side of the transversal formed by the first line and the transversal, measures quantity 5 x minus 8 degrees. An exterior angle formed by the second line and the transversal which is on the right side side of the transversal measures quantity 3 x plus 40 degrees.
Worked Solution
Create a strategy

We can see that the two marked angles form a pair of alternate exterior angles. For there to be a pair of parallel lines, the converse of alternate exterior angles theorem tells us that the two marked angles must be congruent.

Apply the idea

We know that the two marked angles must be congruent, so we can set their measures to be equal and solve for x.

\displaystyle 5x-8\displaystyle =\displaystyle 3x+40Definition of congruent angles
\displaystyle 5x-3x\displaystyle =\displaystyle 40+8Add 8 and subtract 3x from both sides
\displaystyle 2x\displaystyle =\displaystyle 48Combine like terms
\displaystyle x\displaystyle =\displaystyle 24Divide both sides by 2

Therefore, the figure contains a pair of parallel lines when x=24.

Reflect and check

For any other value of x, the equality would not be true and the lines would not be parallel.

Example 3

Determine if the information given is enough to justify the conclusion.

Given: \angle 1 \cong \angle 3

Conclusion: a \parallel c

Worked Solution
Create a strategy

Note that there are no markings on the diagram to indicate any relationship between the lines.

Apply the idea

Given that \angle 1 \cong \angle 3, we could conclude that a \parallel b, using the converse of corresponding angles postulate. However, we do not have enough information to the conclude that a \parallel c. We would need some information involving \angle 4 to draw a conclusion.

Example 4

Consider the following diagram:

Construct a two column proof to prove that \overline{AD} \parallel \overline{BC}.

Worked Solution
Create a strategy

It is given that \overline{AB} \parallel \overline{DC} and \angle ABC is congruent to \angle CDA. Those angles are consecutive interior, this will help us choose which theorems to use in our proof.

Apply the idea
To prove: \overline{AD} \parallel \overline{BC}
StatementsReasons
1.\overleftrightarrow{AB} \parallel \overleftrightarrow{DC}Given
2.\angle ABC \cong \angle CDAGiven
3.\angle ABC and \angle BCD are supplementaryConsecutive interior angles theorem
4.\angle CDA and \angle BCD are supplementarySubstitution
5.\overline{AD} \parallel \overline{BC}Converse of consecutive interior angles theorem
Idea summary

We can use the relationships between angles to choose which theorems we need to prove that lines are parallel.

Constructing parallel lines

Exploration

Click through the slides of the parallel line construction.

Loading interactive...
  1. Describe what is happening in each step.

We can construct a set of parallel lines using dynamic geometry software. Another way to construct parallel lines using a compass and straightedge as follows:

1. Choose a point A through which to construct a line parallel to \overleftrightarrow{BC}.

2. Set the compass width to the distance AC.

3. Construct an arc centered at B with the radius AC.

4. Set the compass width to the distance AB.

5. Construct an arc centered at C with the radius AB.

6. Construct a line through A and the intersection of the two arcs. This line is parallel to \overleftrightarrow{BC}.

Notice, here we did not draw the full circles as was done with the dynamic geometry software. We could have drawn the full circles but it is only necessary to draw enough of the arcs of each circle to make the necessary points of intersection. Drawing a construction this way prevents creating any unnecessary points of intersection that may cause confusion.

Examples

Example 5

Use constructions to construct two parallel lines.

Worked Solution
Create a strategy

We can use a compass and straightedge or technology to construct parallel lines.

Apply the idea

One approach for constructing parallel lines follows:

  1. Use the line segment tool to construct line \overleftrightarrow{AB} and an arbitrary point C not on the line.

  2. Use the point tool to add point D anywhere on line \overleftrightarrow{AB} and use the line tool to draw line \overleftrightarrow{CD}.

  3. Use the point tool to place point E on \overleftrightarrow{CD} closer to C and use the compass tool to construct a circle with radius equal to ED centered at D.

  4. With the compass tool again, construct a circle with radius equal to ED centered at C.

  5. Use the point tool to label the intersection of circle C and \overleftrightarrow{CD} with F and use the point tool to label the intersection of circle D and \overleftrightarrow{AB} with G.

  6. Select the compass tool and create a circle with radius GE centered at F and use the point tool to label the intersection of circle F and circle C with H.

  7. Use the line tool to draw a line through points C and H. This line is parallel to \overleftrightarrow{AB}.

Idea summary

We can use various methods to construct parallel lines including technology and a compass and straightedge.

Outcomes

G.CO.C.9

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

G.CO.D.12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

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