Key takeaway: Parallel lines are lines in the same plane that never meet no matter how far they extend, and on a coordinate plane they always have exactly the same slope.
Grade 10 – GeometryG-CO.C.9Geometry and Measurement
Definition of Parallel Lines
Two parallel lines sit in one plane and stay the same distance apart everywhere, like the two rails of a straight railroad track. They never cross, even if you extend them forever.
Lines that do not touch but are not in the same plane are skew, not parallel. When a third line, a transversal, cuts across a pair of parallel lines, special angle pairs become equal or supplementary. The reverse also works: those angle facts can be used to prove that two lines are parallel.
Formula and notation
The symbol \(\parallel\) means “is parallel to,” so \(\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}\). In the coordinate plane, with slopes \(m_1\) and \(m_2\):
\[ m_1 = m_2 \quad \text{(and different } y\text{-intercepts)} \]
Matching arrowheads drawn on two lines in a diagram show that they are parallel.
Key facts
- Vertical lines are parallel to each other even though their slope is undefined.
- Through a point off a line, exactly one line can be drawn parallel to it.
- A transversal makes equal corresponding angles when the two lines it crosses are parallel.
- Equal slopes with equal y-intercepts describe the same line, not two parallel ones.
Parallel Lines example
Write the equation of the line through \((10, -1)\) that is parallel to \(y = -\frac{2}{5}x + 7\).
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Equal slopes are needed, so the new line has \(m = -\frac{2}{5}\).
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Substitute the point into \(y = mx + b\): \(-1 = -\frac{2}{5}(10) + b = -4 + b\).
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Solve: \(b = 3\).
The line is \(y = -\frac{2}{5}x + 3\); its intercept 3 differs from 7, so the two never meet.
Common mistake
Watch out
Nadia claims \(y = 3x + 2\) and \(y = -3x + 2\) are parallel since they use the same numbers.
Why it is wrong
Parallel lines need the same slope, not the same digits. Here the slopes are 3 and \(-3\), so the lines tilt in opposite directions and cross at \((0, 2)\), their shared intercept.
Related terms and lessons
- Transversal
- Perpendicular Lines
- Same-Side Interior Angles
- Alternate Interior Angles
- Corresponding Angles
- Slope
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