Parallelism Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Parallelism.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Lines in the same plane that never intersect because they maintain a constant distance from each other.

Railroad tracks—they stay exactly the same distance apart and never meet, no matter how far they extend.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Parallel lines have the same slope; the distance between them is constant.

Common stuck point: Parallel lines have equal slopes. In 3D, two lines can be non-intersecting without being parallel (skew lines).

Sense of Study hint: Compare the slopes of both lines. If the slopes are equal and the y-intercepts differ, the lines are parallel.

Worked Examples

Example 1

easy
Line \ell_1 passes through (0, 2) and (4, 6). Write the equation of a line \ell_2 parallel to \ell_1 passing through (1, -3).

Solution

  1. 1
    Step 1: Slope of \ell_1: m_1 = \dfrac{6-2}{4-0} = 1.
  2. 2
    Step 2: Parallel lines have equal slopes, so m_2 = 1.
  3. 3
    Step 3: Point-slope form through (1, -3): y + 3 = 1(x - 1) \Rightarrow y = x - 4.

Answer

y = x - 4
Two distinct lines in the same plane are parallel if and only if they have equal slopes. Here both lines have slope 1 but different y-intercepts (2 and -4), confirming they never intersect.

Example 2

medium
Transversal t crosses parallel lines \ell_1 \parallel \ell_2. If the co-interior (same-side interior) angle at \ell_1 is 65°, find the co-interior angle at \ell_2 and the alternate interior angle at \ell_2.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Are the lines y = 3x + 7 and 6x - 2y = 4 parallel? Justify your answer.

Example 2

hard
Parallelogram PQRS has P(0,0), Q(5,0), R(7,4). Find coordinates of S so that PQ \parallel SR and PS \parallel QR.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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