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Parallel And Perpendicular Lines – Definition, Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Parallel and Perpendicular LinesParallel And Perpendicular Lines – Definition, ExamplesTable of ContentsParallel and Perpendicular Lines Definition of Parallel and Perpendicular Lines

Parallel lines are lines that lie in the same plane and never intersect each other. They always maintain a constant distance between them. When two lines are parallel, we represent this relationship with the symbol "∣∣||∣∣". Parallel lines have the same slope, which means they have the same steepness. The equation of a straight line is represented as y=mx+cy = mx + cy=mx+c, where "mmm" is the slope and "ccc" is the y-intercept. Two lines are parallel when they have equal slopes.

Perpendicular lines are lines that intersect each other at a right angle (90°). We use the symbol "⊥\bot⊥" to represent perpendicular lines. Unlike parallel lines, perpendicular lines have slopes that are negative reciprocals of each other. This means the product of their slopes equals -1, which can be expressed mathematically as m1×m2=−1m_1 \times m_2 = -1m1​×m2​=−1, where m1m_1m1​ and m2m_2m2​ are the slopes of the two perpendicular lines.

Examples of Parallel and Perpendicular Lines Example 1: Finding the Slope of a Parallel Line Problem:

If the slope of one of the two parallel lines is 5, then what will be the slope of the other parallel line?

Step-by-step solution:

Step 1, Identify what we know. We're given that k1=5k_1 = 5k1​=5 for the first line.

Step 2, Recall that parallel lines have equal slopes. This means k1=k2k_1 = k_2k1​=k2​ for any two parallel lines.

Step 3, Apply this rule to find our answer. Since k1=k2k_1 = k_2k1​=k2​, we can say that k2=5k_2 = 5k2​=5. Parallel Lines

Example 2: Checking if Lines are Perpendicular Problem:

Find the slopes of the lines 5x+2y−6=05x + 2y - 6 = 05x+2y−6=0 and −2x+5y+3=0-2x + 5y + 3 = 0−2x+5y+3=0. Also, which types of lines are they?

Step-by-step solution:

Step 1, Rearrange the first equation to find the slope. Let's solve for yyy:

5x+2y−6=05x + 2y - 6 = 05x+2y−6=0 2y=−5x+62y = -5x + 62y=−5x+6 y=−52x+3y = \frac{-5}{2}x + 3y=2−5​x+3 So, k1=−52k_1 = \frac{-5}{2}k1​=2−5​

Step 2, Rearrange the second equation to find its slope:

−2x+5y+3=0-2x + 5y + 3 = 0−2x+5y+3=0 5y=2x−35y = 2x - 35y=2x−3 y=25x−35y = \frac{2}{5}x - \frac{3}{5}y=52​x−53​ So, k2=25k_2 = \frac{2}{5}k2​=52​

Step 3, Check if the product of the slopes equals -1:

k1×k2=−52×25=−1k_1 \times k_2 = \frac{-5}{2} \times \frac{2}{5} = -1k1​×k2​=2−5​×52​=−1

Step 4, Make a conclusion. Since the product of the slopes is -1, these lines are perpendicular.

Checking if Lines are Perpendicular

Example 3: Writing the Equation of a Parallel Line Problem:

If two lines are parallel to each other and equation of one line is y=−7x+3y = -7x + 3y=−7x+3 and the point on the other line is (2,−5)(2,-5)(2,−5), then what will be the equation of the other line?

Step-by-step solution:

Step 1, Find the slope of the first line. From y=−7x+3y = -7x + 3y=−7x+3, we can see that m1=−7m_1 = -7m1​=−7.

Step 2, Since the lines are parallel, we know the second line must have the same slope. So m2=m1=−7m_2 = m_1 = -7m2​=m1​=−7.

Step 3, Use the point-slope form to find the equation. We know a point (2,−5)(2,-5)(2,−5) on the line and the slope m2=−7m_2 = -7m2​=−7.

Step 4, Substitute these values into the equation y=mx+cy = mx + cy=mx+c:

−5=−7×2+c-5 = -7 \times 2 + c−5=−7×2+c −5=−14+c-5 = -14 + c−5=−14+c c=−5+14=9c = -5 + 14 = 9c=−5+14=9

Step 5, Write the final equation of the parallel line: y=−7x+9y = -7x + 9y=−7x+9

Parallel Lines

Comments(6)CCricketFollowerVioletNovember 6, 2025This glossary page is a lifesaver! It's made explaining parallel and perpendicular lines to my students so much easier. Thanks!

MCMs. CarterSeptember 17, 2025This explanation of parallel and perpendicular lines was a lifesaver for my 7th grader’s homework! The examples made it so easy to understand, and the slope tips were super helpful. Thank you!

MCMs. CarterSeptember 10, 2025This page was a game-changer for my kids! The clear examples of parallel and perpendicular lines made homework so much easier. I’ve bookmarked it for future geometry lessons!

MCMs. CarterAugust 27, 2025I used the Parallel And Perpendicular Lines examples from this page to help my son with his geometry homework. The clear explanations about slopes made it so much easier for him to understand. Highly recommend!

MCMs. CarterAugust 20, 2025I’ve been using this page to help my kids understand parallel and perpendicular lines for their math homework. The examples and step-by-step solutions make it so much easier to explain—thanks for such a clear resource!

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