Parallel Lines

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Parallel Lines: meaning, method and solved example (Geometry) – Maths PDF

In short: Parallel lines sit in one plane and never meet, staying the same distance apart everywhere, and on a coordinate grid they have equal slopes.

GeometryGeometryG-CO.C.9

What is Parallel Lines?

Parallel lines lie in a single plane and never cross, however far you extend them. The two rails of a straight railroad track are the classic picture. The symbol \(\ell \parallel m\) reads line \(\ell\) is parallel to line \(m\).

Slice such a pair with a third line, and the eight angles that appear fall into two sizes: matching positions agree, and inside angles on the same side add to \(180^{\circ}\). On a graph, non-vertical ones share one slope but differ in y-intercept, such as \(y = 2x + 1\) and \(y = 2x – 6\).

How to spot it

  • A figure puts matching arrowheads in the middle of two lines.
  • The wording says parallel, or the symbol \(\parallel\) stands between two line names.
  • A third line slices across two others, and angle measures at matching corners are listed.
  • Two equations show the same coefficient of x but different constants.

The method

  1. Look for evidence: matching arrowheads, equal slopes or equal corresponding angles.
  2. With a transversal, set a congruent pair equal, or make a same-side pair add to \(180^{\circ}\), then solve.
  3. On a graph, rewrite each equation as \(y = mx + b\) and compare the values of \(m\).
  4. To write a new one through a point, keep the slope and solve for the new intercept.
65°65°line lline ml is parallel to m: corresponding angles are equal
A transversal makes equal corresponding angles with two parallel lines.

Parallel Lines: solved example

Problem

Find the equation of the line through \((6, 1)\) that is parallel to \(y = \frac{2}{3}x – 5\).

Solution

Parallel lines share a slope, so the new line also has \(m = \frac{2}{3}\). Use point-slope form: \(y – 1 = \frac{2}{3}(x – 6)\).

Expand: \(y – 1 = \frac{2}{3}x – 4\), so \(y = \frac{2}{3}x – 3\). Check at \(x = 6\): \(\frac{2}{3}(6) – 3 = 1\). The line is \(y = \frac{2}{3}x – 3\).

Try it

Question 1

Are the graphs of \(y = 3x + 7\) and \(6x – 2y = 5\) parallel?

Show the answer

The second becomes \(y = 3x – 2.5\). The slopes match (3) and the intercepts differ (7 and \(-2.5\)), so they are parallel.

Question 2

A road crosses two parallel streets. At the matching corners the angles read \((5x + 10)^{\circ}\) and \(75^{\circ}\). Find \(x\).

Show the answer

Matching corners are equal: \(5x + 10 = 75\), so \(5x = 65\) and \(x = 13\).

Parallel Lines vs. Perpendicular Lines

Parallel pairs never meet and share one slope, such as 2 and 2. Perpendicular pairs meet at a right angle and have slopes whose product is \(-1\), such as 2 and \(-\frac{1}{2}\). Same slope means parallel; negative reciprocal slopes mean perpendicular.

Perpendicular Lines

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