Parallel & Perpendicular Slopes

📖 The Two Rules

Every question about parallel and perpendicular lines comes down to one thing: comparing slopes. There are only two rules to remember.

Rule 1 — Parallel Lines

Same slope = Parallel

If Line A has slope 34 and Line B has slope 34 — they are parallel. They run in exactly the same direction and never cross.

Rule 2 — Perpendicular Lines

Flip + change sign = Perpendicular

If Line A has slope 23 — flip it to 32, then change the sign to −32. A line with slope −32 is perpendicular to Line A.

That's really it. Every parallel and perpendicular question on the GED is asking you to do one of these two things — match a slope, or flip and change the sign.

Parallel lines — same slope

Perpendicular lines — right angle

➡️ Parallel Lines — Same Slope

Parallel lines run in the same direction. They never cross. On a graph they look like they're traveling side by side.

To check if two lines are parallel: find both slopes and compare them. If the slopes are identical — including the sign — the lines are parallel.

m₁ = m₂

Same slope → Parallel lines

Example A: Line A has slope 34 Line B has slope 34

Compare: same numerator, same denominator, same sign ✓

The lines are parallel

Example B: y = 5x + 2 and y = 5x − 7

Line A slope = 5 Line B slope = 5 (read the number in front of x)

Same slope → the lines are parallel (different y-intercepts just shift them up/down)

Example C — Watch out: Line A slope = 23 Line B slope = −23

Same numbers but opposite signs → NOT the same slope

These lines are NOT parallel — they are neither parallel nor perpendicular

Watch out: m = 3 and m = −3 are NOT the same slope. Parallel needs an exact match — same number AND same sign.

↗️↘️ Perpendicular Lines — Flip & Change Sign

Perpendicular lines meet at a perfect right angle (90°). Their slopes are opposite reciprocals — but you don't need to memorize that term. Just remember the two steps:

Step 1: Flip the fraction Step 2: Change the sign

Do BOTH steps — missing either one gives the wrong answer

Example A: Slope = 23

Flip → 32 Change sign → −32

Perpendicular slope = −32

Example B: Slope = −4 (write as −41 first)

Flip → −14 Change sign → 14

Perpendicular slope = 14

Example C: Slope = −52

Flip → −25 Change sign → 25

Perpendicular slope = 25

Most common mistake: Flipping the fraction but forgetting to change the sign. You must do BOTH. The perpendicular of 3 is −13 — not 13.

🔢 Every Whole Number Is Secretly a Fraction

This is one of the most important ideas in this whole lesson — and one of the most skipped. Before you can flip a whole-number slope to find the perpendicular, you have to see it as a fraction first.

Any whole number can be written as itself over 1. That doesn't change the value — it just makes the fraction visible so you know what to flip.

Why it works — the visual

Writing 4 as 41 doesn't change anything — 4 ÷ 1 is still 4. But now you can see the top and bottom clearly, which means you can flip it to get 14.

Every whole number = that number over 1

4 = 4/1 · −3 = −3/1 · 7 = 7/1 · −10 = −10/1

Example A: Find the perpendicular slope of m = 4

Step 1: Write as a fraction → 41

Step 2: Flip it → 14

Step 3: Change the sign → −14

Perpendicular slope = −14

Example B: Find the perpendicular slope of m = −3

Step 1: Write as a fraction → −31

Step 2: Flip it → −13

Step 3: Change the sign → 13 (negative becomes positive)

Perpendicular slope = 13

Quick check: Multiply the two slopes together. If they are perpendicular, the answer is always −1.
4 × (−14) = −1 ✓ −3 × 13 = −1 ✓

🥷 The Ninja 1 — The Invisible Slope

Here is a trick the GED uses all the time — and it catches a lot of students off guard. When you see an equation like y = x + 3 or y = −x + 5, there is no number written in front of x. That missing number is not zero. It is an invisible 1 (or −1).

Why "ninja"? Because it's hiding. It's there doing its job but you can't see it. In algebra, writing 1 in front of a variable is optional — mathematicians drop it because it doesn't change the value. But on a test, that hidden 1 trips up students who forget it's there.

y = x + 3 means y = 1x + 3

Slope = 1

The 1 is invisible but it's there. This line has a slope of 1 — it rises 1 unit for every 1 unit to the right. It's a 45° angle line.

y = −x + 5 means y = −1x + 5

Slope = −1

The −1 is invisible but it's there. This line falls 1 unit for every 1 unit to the right. It's the mirror image of slope 1.

The Ninja 1 in action

You see

y = x + 3

What it means

y = 1x + 3

The slope

m = 1

Perpendicular of y = x + 3 (slope = 1)

Step 1: Write as a fraction → 11

Step 2: Flip it → 11 (flipping 1/1 gives 1/1 — same thing!)

Step 3: Change the sign → −11 = −1

Perpendicular slope = −1 (and the perpendicular of −1 is 1)

Parallel to y = x + 3 (slope = 1)

A parallel line also has slope = 1.

Example: y = x − 7 is parallel to y = x + 3. (Same slope, different y-intercept.)

Any line written as y = x + (any number) is parallel to y = x + 3

This shows up constantly on the GED. When you see y = x or y = −x, don't panic — just remember the ninja 1 is hiding there. Treat it exactly like any other whole number slope. The only difference is that you can't see it written.

🔄 When the Equation Isn't in y = mx + b Form

Sometimes an equation is written in standard form like 2x + y = 6. You can't read the slope directly — but it only takes one step to fix that.

Example: 2x + y = 6

Subtract 2x from both sides: y = −2x + 6

Slope = −2

Example: x + 2y = 10

Subtract x: 2y = −x + 10

Divide by 2: y = −12x + 5

Slope = −12

The goal: Get y by itself on the left side. Once you have y = mx + b, the slope is right there in front of x.

⚠️ Common Mistakes

Mistake 1

Flipping but not changing the sign

Perpendicular of 3 is −13, not 13. You must do BOTH steps.

Mistake 2

Changing the sign but not flipping

Perpendicular of 25 is −52, not −25. Flip the fraction too.

Mistake 3

Confusing 3 and −3 as parallel

m = 3 and m = −3 are not the same. Parallel requires an exact match.

Mistake 4

Comparing y-intercepts instead of slopes

Whether lines are parallel or perpendicular depends ONLY on slopes — never on y-intercepts.

Parallel &Perpendicular Slopes