Parallel lines

Parallel lines are lines in the same plane that never meet, however far they are extended. They stay the same distance apart all the way along, like the rails of a train track. On a graph, parallel lines have the same slope. The symbol for parallel is ∥, so AB ∥ CD means line AB is parallel to line CD, and diagrams mark parallel lines with matching arrowheads.

What are parallel lines?

Two lines are parallel when they lie in the same flat surface and never cross. Three things are true of every pair of parallel lines:

• They never intersect, even if you extend them forever.
• They are always the same distance apart.
• They point in the same direction, so on a coordinate grid they have equal slopes.

Everyday examples include the lines on notebook paper, the opposite edges of a ruler or a door, the strings of a guitar, and the lanes on a running track. Lines in different planes that never meet, like a line along a ceiling and a line across the floor going another way, are called skew lines, not parallel lines.

Parallel lines and transversals

A transversal is a line that crosses two or more other lines. When it crosses two parallel lines, it makes eight angles, and they come in just two sizes: four of one size and four of the other, adding up to 180°.

In the figure, the 65° angle at the lower crossing and the 65° angle up and to the right at the upper crossing are corresponding. The 65° angle to the left below the upper line is alternate interior to the lower 65°. The 115° and the lower 65° sit side by side on a straight line, so they are supplementary. Each pair has its own guide: corresponding angles, alternate interior angles, vertical angles and supplementary angles.

In the US, grade 8 students explain these angle facts informally, and high school geometry proves them, under the Common Core standards.

Example 1: alternate interior angles

Two parallel lines are cut by a transversal. A pair of alternate interior angles measure (3x + 10)° and (5x − 30)°. Find x and the angles.

Both angles are 3(20) + 10 = 70°.

Example 2: same-side interior angles

Same-side interior angles measure (2x + 15)° and (4x − 15)°. They add up to 180°:

The angles are 75° and 105°.

How to prove lines are parallel

The angle facts also work backward. If a transversal crosses two lines and any one of these is true, the lines are parallel:

• A pair of corresponding angles is equal.
• A pair of alternate interior angles is equal.
• A pair of alternate exterior angles is equal.
• A pair of same-side interior angles adds up to 180°.

So if a transversal makes a 72° angle with one line and a 72° corresponding angle with another, those two lines must be parallel. If the angles were 72° and 73°, the lines would eventually meet.

Check you can spot a parallel pair.

Slopes of parallel lines

On a coordinate grid, two non-vertical lines are parallel exactly when they have the same slope and different y-intercepts. In y = mx + b form, compare the values of m.

Both lines rise 2 for every 1 across, so they never get closer together. Under the Common Core standards, high school students prove this slope test and use it to solve problems.

Example 3: are these lines parallel?

• y = 3x − 4 and y = 3x + 7: both slopes are 3 and the intercepts differ, so yes.
• y = 3x − 4 and y = −3x + 7: the slopes are 3 and −3, so no. They cross.
• 2x + y = 5 and y = −2x + 1: rewrite the first as y = −2x + 5. Both slopes are −2, so yes.

Example 4: a parallel line through a point

Find the line parallel to y = 2x + 1 that passes through (4, 1).

A parallel line has the same slope, 2. Use point-slope form with the point (4, 1):

Check: 2(4) − 7 = 1. The point-slope form guide has more examples, and you can graph both lines in the Linear Graph calculator.

Finding slopes is the key step. Try one from the Graphing Functions course.

Parallel, perpendicular and intersecting lines

Vertical lines such as x = 2 and x = 5 are parallel to each other even though their slopes are undefined. Horizontal lines such as y = 1 and y = 4 are parallel with slope 0. More on the right-angle case is in the perpendicular lines guide.

Common mistakes

• Thinking lines that don't meet on the page are parallel. Extend them: if they would cross somewhere, they are not parallel.
• Using parallel angle rules when the lines are not parallel. Check for arrow marks or a statement before using them.
• Calling same-side interior angles equal. They add up to 180°.
• Comparing y-intercepts instead of slopes. Parallel lines share a slope, not an intercept. Same slope and same intercept means the same line.
• Forgetting to rearrange first. Put each equation in y = mx + b form before comparing slopes.

Practice with a short timed paper

Parallel lines, angle pairs and slopes come up all through geometry, algebra and the SAT. The daily Geometry, Algebra 2 & SAT Math and Pre-Algebra & Algebra 1 papers are short timed practice tests of 8 questions, with a new one every day.

Read next

Maths · GeometryPerpendicular lines: meaning, slopes and equationsWhat perpendicular means, how to spot perpendicular lines from their slopes, and how to find the equation of a perpendicular line, with worked examples.