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Parallel and Perpendicular Lines

Use slopes to recognize parallel and perpendicular lines and to write their equations.

Practice this skill 25 min Parallel and perpendicular lines Leer en español
Key idea

What you'll learn

Two different lines in a plane either cross or never meet. Slope tells you which — and whether they cross at a perfect right angle.

You will learn to:

  • recognize parallel lines: same slope, never meet;
  • recognize perpendicular lines: slopes that are negative reciprocals, meeting at a 90^\circ angle;
  • find the slope of a line from any form — y = mx + b, Ax + By = C, two points, or a graph;
  • write the equation of a line parallel or perpendicular to a given line through a given point;
  • use slopes to check shapes (rectangles, right triangles) and solve for an unknown.

Why it matters on the GED: You will see items like "Which line is perpendicular to y = -2x + 7?" or "Write the equation of the line parallel to y = \tfrac{1}{2}x + 3 through (4, 5)." Geometry items may ask whether four points make a rectangle. The slope rules for parallel and perpendicular lines are not on the formula sheet — you must remember them.

Real life: Builders, surveyors, and city planners use the same idea: streets that never meet are parallel; a road that crosses another "at a right angle" is perpendicular.

Words to know

  • Slope (m) — steepness, \dfrac{\text{rise}}{\text{run}}. In y = mx + b it is the number in front of x.
  • Parallel lines — lines in the same plane that never meet. Symbol: \parallel. Example: y = 2x + 1 and y = 2x - 3.
  • Perpendicular lines — lines that meet at a right angle (90^\circ). Symbol: \perp. Example: y = 2x + 1 and y = -\tfrac{1}{2}x + 2.
  • Right angle — a 90^\circ angle, like the corner of a sheet of paper; marked with a small square.
  • Reciprocal — flip a fraction upside down. The reciprocal of \tfrac{2}{3} is \tfrac{3}{2}; of 4 (which is \tfrac{4}{1}) it is \tfrac{1}{4}.
  • Negative (opposite) reciprocal — flip the fraction and change the sign. Of \tfrac{2}{3}: -\tfrac{3}{2}. Of -5: \tfrac{1}{5}.
  • Standard form — Ax + By = C. Example: 2x + 3y = 6. Its slope is -\tfrac{A}{B}.
  • Coinciding lines — the same line written twice (same slope and same intercept). They are not "parallel"; they overlap.
  • Horizontal line — slope 0, y = c. Vertical line — undefined slope, x = c.

All Math words in the glossary

Parallel lines: same slope

Rule 1: Two non-vertical lines are parallel exactly when they have the same slope and different y-intercepts.

Line 1Line 2SlopesRelationship
y = 3x - 5y = 3x + 23 and 3parallel
y = -\tfrac{1}{2}x + 4y = -\tfrac{1}{2}xequalparallel
y = 3x + 2y = -3x + 23 and -3not parallel (they cross at (0, 2))
y = 2x + 12y = 4x + 22 and 2, same bthe same line

Rule 2: The sign matters. 3 and -3 are different slopes: one line rises, the other falls.

Rule 3: The y-intercept does not make lines parallel. Two lines with the same b meet at that point.

Rule 4: All vertical lines are parallel to each other (such as x = 2 and x = -5), and all horizontal lines are parallel to each other (such as y = 1 and y = 4).

Why it works: Lines with equal slope rise the same amount for every step to the right, so the gap between them never changes — they can never meet.

Coordinate plane with two parallel lines. Blue line y = 3x + 2 and red line y = 3x - 5 both have slope 3: each green slope triangle goes right 1 and up 3. The lines keep the same vertical gap of 7 and never meet.−4−3−212345−9−8−4−3−2−134670xyright 1up 3right 1up 3(0, 2)(0, −5)y = 3x + 2y = 3x − 5

Perpendicular lines: negative reciprocal slopes

Rule 5: Two non-vertical lines are perpendicular exactly when their slopes are negative reciprocals — flip the fraction and change the sign. Equivalently, the product of the slopes is -1:

m_1 \cdot m_2 = -1
Slope of a linePerpendicular slopeProduct
2-\tfrac{1}{2}2 \cdot (-\tfrac{1}{2}) = -1
-3\tfrac{1}{3}-1
\tfrac{3}{4}-\tfrac{4}{3}-1
-\tfrac{2}{5}\tfrac{5}{2}-1
1-1-1
0 (horizontal)undefined (vertical)—
Coordinate plane with two perpendicular lines through the origin, marked with a right-angle square. Blue line y = (3/4)x: its green slope triangle goes right 4 and up 3. Red line y = -(4/3)x: its slope triangle goes right 3 and down 4. The slopes 3/4 and -4/3 are negative reciprocals: (3/4)(-4/3) = -1.−6−5−4−3456−5−4−3−2−112345xyright 4up 3right 3down 4y = ¾xy = −4/3 x

Rule 6: You must do both steps — flip AND change the sign.

  • Only flipping 2 gives \tfrac{1}{2} ✗ (product 1).
  • Only changing the sign gives -2 ✗ (product -4).
  • Both: -\tfrac{1}{2} ✓

Rule 7: Horizontal and vertical lines are perpendicular. y = 3 and x = -1 meet at a right angle, even though the product rule can't be used (a vertical line has no slope).

Coordinate plane with the horizontal line y = 3 in blue and the vertical line x = -1 in red. They cross at (-1, 3), where a small square marks a right angle: horizontal and vertical lines are perpendicular.−4−3−21234−2−112450xy(−1, 3)y = 3x = −1

Decimals: a slope of 0.25 = \tfrac{1}{4} has perpendicular slope -4. A slope of -1.5 = -\tfrac{3}{2} has perpendicular slope \tfrac{2}{3}.

Coordinate plane from -5 to 5 on both axes with three lines. Solid blue line y = 2x + 1 and dashed blue line y = 2x - 3 both have slope 2, so they are parallel and never meet. Black line y = -(1/2)x + 2 has slope -1/2 and crosses y = 2x + 1 at (0.4, 1.8), where a small square marks a right angle: the lines are perpendicular.-5-4-3-2-112345-5-4-3-2-1123450xyy = 2x + 1y = 2x − 3y = −(1/2)x + 2

y = 2x + 1 and y = 2x - 3 both have slope 2: they are parallel. y = -\tfrac{1}{2}x + 2 has slope -\tfrac{1}{2}, and 2 \cdot (-\tfrac{1}{2}) = -1, so it is perpendicular to both blue lines — note the right-angle mark.

Step one is always: find the slope

The GED rarely hands you the slope directly. Get it first:

  • From y = mx + b: the number in front of x. In y = -\tfrac{3}{4}x + 2, m = -\tfrac{3}{4}.
  • From standard form Ax + By = C: solve for y. For 2x + 3y = 6: 3y = -2x + 6, so y = -\tfrac{2}{3}x + 2 and m = -\tfrac{2}{3}. (Shortcut: m = -\tfrac{A}{B} = -\tfrac{2}{3}.) The slope is not 2 — the coefficient of x only equals the slope once y is alone.
  • From two points: m = \dfrac{y_2 - y_1}{x_2 - x_1}. Through (1, 2) and (3, 8): m = \tfrac{6}{2} = 3.
  • From a graph: pick two grid points on the line and count rise over run.

Then write the new line using the method from the earlier lessons:

  1. Find the slope of the given line.
  2. Parallel → use the same slope. Perpendicular → use the negative reciprocal.
  3. Put the new slope and the given point into y = mx + b and solve for b.

Example: parallel to y = 4x - 1 through (2, 3): m = 4; 3 = 4(2) + b, so b = -5 → y = 4x - 5.

Example: perpendicular to y = 4x - 1 through (2, 3): m = -\tfrac{1}{4}; 3 = -\tfrac{1}{4}(2) + b = -\tfrac{1}{2} + b, so b = \tfrac{7}{2} → y = -\tfrac{1}{4}x + \tfrac{7}{2}.

Geometry with slopes:

  • A parallelogram has both pairs of opposite sides parallel (equal slopes).
  • A rectangle is a parallelogram whose neighboring sides are perpendicular (slopes multiply to -1).
  • A triangle has a right angle at a vertex if the two sides meeting there are perpendicular.

Finding an unknown: if a line through (2, 5) and (6, k) must be perpendicular to y = -2x + 1, its slope must be \tfrac{1}{2}: \dfrac{k - 5}{4} = \dfrac{1}{2}, so k = 7.

Coordinate plane with the red line y = -2x + 1 and the blue line through (2, 5) and (6, 7), which has slope 1/2. A green slope triangle goes right 4 and up 2 from (2, 5) to (6, 7), so k = 7. The lines cross at (-1.2, 3.4) at a right angle, marked with a small square, because 2/4 = 1/2 and (1/2)(-2) = -1.−4−3−2−12345678−3−2−113567890xyrun 4rise 2(2, 5)(6, k) = (6, 7)y = −2x + 1slope ½
Formula

The rules (NOT on the GED formula sheet — memorize them)

\text{Parallel: } m_1 = m_2 \qquad\qquad \text{Perpendicular: } m_2 = -\frac{1}{m_1} \;\;\text{(so } m_1 \cdot m_2 = -1\text{)}
  • m_1 — slope of the first line; m_2 — slope of the second line
  • Standard form Ax + By = C has slope m = -\dfrac{A}{B}

The formula sheet does list the slope formula m = \dfrac{y_2 - y_1}{x_2 - x_1}, slope-intercept form y = mx + b, and point-slope form y - y_1 = m(x - x_1). The parallel and perpendicular rules above are not on it. Memory hook: parallel = "twins" (identical slopes); perpendicular = "flip and switch" (flip the fraction, switch the sign).

Worked example

Example 1 (easy): a parallel line through a point

Problem: Write the equation of the line parallel to y = 4x - 1 that passes through (2, 3).

  1. Slope of the given line: 4. Parallel → new slope is also 4.
  2. Substitute (2, 3) into y = 4x + b: 3 = 4(2) + b = 8 + b.
  3. Solve: b = -5.
Coordinate plane with the given line y = 4x - 1 in blue and the parallel line y = 4x - 5 in red through the point (2, 3). A green slope triangle on the red line goes right 1 and up 4: both lines have the same slope 4. The y-intercepts are (0, -1) and (0, -5).−4−3−2−12345−4123456780xy1up 4(2, 3)(0, −1)(0, −5)y = 4x − 1y = 4x − 5

Answer: y = 4x - 5

Check: 4(2) - 5 = 3 ✓ and slopes 4 = 4 with different intercepts (-5 \ne -1) ✓

Worked example

Example 2 (medium): a perpendicular line through a point

Problem: Write the equation of the line perpendicular to y = -\tfrac{3}{5}x + 2 that passes through (3, 1).

  1. Given slope: -\tfrac{3}{5}.
  2. Flip and switch: perpendicular slope = +\tfrac{5}{3}.
  3. Substitute (3, 1): 1 = \tfrac{5}{3}(3) + b = 5 + b, so b = -4.
Coordinate plane with the given line y = -(3/5)x + 2 in blue and the perpendicular line y = (5/3)x - 4 in red through (3, 1). A green slope triangle on the blue line goes right 5 and down 3; one on the red line goes right 3 and up 5. The lines cross near (2.6, 0.4) at a right angle, marked with a small square.−4−3−2−1135678−4−3−2−1460xyright 5down 3right 3up 5(3, 1)(0, 2)y = −⅗x + 2y = 5/3 x − 4

Answer: y = \tfrac{5}{3}x - 4

Check: -\tfrac{3}{5} \cdot \tfrac{5}{3} = -1 ✓ and \tfrac{5}{3}(3) - 4 = 1 ✓

Worked example

Example 3 (medium): the given line is in standard form

Problem: Find the equation of the line perpendicular to 4x - 2y = 6 that passes through (-4, 1).

  1. Solve for y: -2y = -4x + 6; divide by -2: y = 2x - 3. Slope = 2.
  2. Perpendicular slope: -\tfrac{1}{2}.
  3. Substitute (-4, 1): 1 = -\tfrac{1}{2}(-4) + b = 2 + b, so b = -1.
Coordinate plane with the given line 4x - 2y = 6, which is y = 2x - 3, in blue, and the perpendicular line y = -(1/2)x - 1 in red through (-4, 1). Slope triangles: right 1 and up 2 on the blue line; right 2 and down 1 on the red line. The lines cross at (0.8, -1.4) at a right angle, marked with a small square.−6−5−4−3−22345−6−5−3−22340xyright 1up 2right 2down 1(−4, 1)y = 2x − 3(4x − 2y = 6)y = −½x − 1

Answer: y = -\tfrac{1}{2}x - 1

Check: 2 \cdot (-\tfrac{1}{2}) = -1 ✓ and -\tfrac{1}{2}(-4) - 1 = 2 - 1 = 1 ✓

Worked example

Example 4 (word problem): a new road at a right angle

Problem: On a town map (units are blocks), a straight railroad passes through (-2, 1) and (4, 4). The town wants a new road that meets the railroad at a right angle at the station, (4, 4). Write the road's equation, and find where it meets Highway 1, which runs along the x-axis.

  1. Railroad slope: \dfrac{4 - 1}{4 - (-2)} = \dfrac{3}{6} = \dfrac{1}{2}.
  2. Road is perpendicular: slope = -2.
  3. Through (4, 4): 4 = -2(4) + b = -8 + b, so b = 12. Road: y = -2x + 12.
  4. Highway 1 is y = 0: 0 = -2x + 12, so x = 6.
Town map grid in blocks. The blue railroad passes through (-2, 1) and (4, 4), slope 3/6 = 1/2 (green triangle right 6, up 3). The red road y = -2x + 12 meets the railroad at the station (4, 4) at a right angle, marked with a small square, and meets Highway 1 (the x-axis) at (6, 0).−3−2−11234578−13456780xyright 6up 3(−2, 1)station (4, 4)(6, 0)Highway 1 (y = 0)railroadroad y = −2x + 12

Answer: Road: y = -2x + 12; it meets Highway 1 at (6, 0).

Check: \tfrac{1}{2} \cdot (-2) = -1 ✓ and -2(4) + 12 = 4 ✓

Worked example

Example 5 (GED-hard, work backwards): a right-angled garden corner

Problem: A landscaper marks the corners of a triangular garden on a grid (in feet) at P(1, 2), Q(4, 6) and R(8, k). She wants the corner at Q to be a right angle. Find k.

  1. Side PQ: slope \dfrac{6 - 2}{4 - 1} = \dfrac{4}{3}.
  2. For a right angle at Q, side QR must be perpendicular to PQ: slope -\tfrac{3}{4}.
  3. Slope of QR from (4, 6) to (8, k): \dfrac{k - 6}{8 - 4} = \dfrac{k - 6}{4}.
  4. Set equal: \dfrac{k - 6}{4} = -\dfrac{3}{4}, so k - 6 = -3 and k = 3.
Grid in feet with the triangular garden P(1, 2), Q(4, 6), R(8, 3). Side PQ has slope 4/3 (green triangle right 3, up 4) and side QR has slope -3/4 (right 4, down 3). The corner at Q is a right angle, marked with a small square, because (4/3)(-3/4) = -1.0123456789012345678right 3up 4right 4down 3P(1, 2)Q(4, 6)R(8, k) = (8, 3)slope 4/3slope −3/4

Answer: k = 3, so R = (8, 3).

Check: Slope QR = \dfrac{3 - 6}{4} = -\dfrac{3}{4}, and \dfrac{4}{3} \cdot \left(-\dfrac{3}{4}\right) = -1 ✓

Calculator tip

Calculator tip: TI-30XS MultiView

Most parallel/perpendicular questions are about recognizing slopes, which is faster without a calculator. Use it for the arithmetic around them:

  • Checking perpendicularity with decimals: multiply the slopes. For 1.25 and -0.8: 1.25 × (−) 0.8 enter → −1, so they are perpendicular.
  • Negative reciprocal of a decimal: type (−) 1 ÷ 1.25 enter → −0.8. Press toggle to see it as a fraction, -\tfrac{4}{5}.
  • Fractions: use the n/d key for slopes such as \tfrac{5}{3}, and toggle to switch between fraction and decimal answers.
  • Finding b: compute y_1 - m x_1, putting a negative slope or coordinate in parentheses with the (−) key.

When NOT to use it: flipping \tfrac{2}{3} to -\tfrac{3}{2} is a mental step — typing it takes longer and invites mistakes.

Common mistake

Common mistakes

  • Mixing up the rules. Parallel = same slope; perpendicular = negative reciprocal. Using -\tfrac{1}{2} for a line parallel to y = 2x is wrong.
  • Doing only half of "flip and switch." Perpendicular to slope 3: not \tfrac{1}{3} (only flipped), not -3 (only sign changed); it is -\tfrac{1}{3}.
  • Reading the slope from standard form without solving for y. In 2x + 3y = 6, the slope is -\tfrac{2}{3}, not 2.
  • Thinking equal y-intercepts make lines parallel. y = 2x + 5 and y = -x + 5 share b = 5 but meet at (0, 5).
  • Copying the old intercept. A parallel line through a new point needs a new b; reusing the old one gives back the original line.
  • "Opposite signs" ≠ perpendicular. Slopes 2 and -2 are not perpendicular: 2 \cdot (-2) = -4, not -1.
Test strategy

GED test strategy

  • Find the target slope before reading the choices. Write it on your erasable note board: "need m = -\tfrac{1}{2}." Then scan the choices for that slope only.
  • Expect one distractor for each trap: the parallel slope, the reciprocal without the sign change, the sign change without the reciprocal. Knowing this makes elimination quick.
  • Rewrite any choice not in y = mx + b form before comparing slopes.
  • Test the point last. Among choices with the right slope, plug in the given point to pick the one that passes through it.
  • Use the product test for confidence: if m_1 \cdot m_2 = -1, they are perpendicular. Equal slopes with different intercepts → parallel.
  • Sketch. A quick sketch shows whether a line should rise or fall and roughly where it crosses the y-axis.

Check yourself

Parallel and perpendicular lines

Which equation represents a line that is parallel to the line y = 3x - 5?

Answer choices
Hint

Parallel lines rise or fall at exactly the same rate. Which part of y = mx + b describes that?

Check yourself

Parallel and perpendicular linesFill in

A line has a slope of \tfrac{4}{5}.

What is the slope of a line perpendicular to it? Enter a decimal or a fraction.

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Perpendicular slopes are negative reciprocals: two changes are needed.

Summary

  • Parallel lines have equal slopes (and different intercepts). All vertical lines are parallel; so are all horizontal lines.
  • Perpendicular lines have negative reciprocal slopes: flip the fraction and switch the sign; the slopes multiply to -1. Horizontal ⟂ vertical.
  • Always find the slope first — from y = mx + b, from Ax + By = C (slope -\tfrac{A}{B}), from two points, or from a graph.
  • To write a parallel or perpendicular line through a point: choose the new slope, then plug in the point and solve for b.
  • These two rules are not on the formula sheet — memorize "twins" and "flip and switch."
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