Proving Slope Criteria for Lines

Lesson 1 of 2: Why the Criteria Are True

In this lesson:

• Prove parallel lines have equal slopes
• Prove perpendicular lines have slopes with product −1

Learning Objectives

By the end of this lesson, you should be able to:

1. Prove two non-vertical lines are parallel if and only if they have equal slopes
2. Prove two non-vertical lines are perpendicular if and only if the product of their slopes is

What Does Slope Actually Measure?

You have computed slope as since 8th grade.

• Slope measures steepness and direction
• Two lines pointing in the same direction → same slope
• Parallel lines point in the same direction

So parallel lines must have the same slope — but can we prove it?

Slope Triangles on Parallel Lines

Both triangles have the same rise-over-run — so both lines have the same slope.

Translation Preserves the Slope Triangle

• Translation maps line onto line
• The same translation maps the slope triangle of onto a congruent triangle on
• Congruent triangles → equal rise and equal run
• Therefore

Equal Angles Also Prove Equal Slopes

A line making angle with the positive -axis has slope:

• Parallel lines make equal angles with any transversal — including horizontal lines
• So , which means
• Therefore

The Parallel Criterion Is a Biconditional

Two distinct non-vertical lines are parallel if and only if they have equal slopes.

• Parallel → equal slopes: translation maps one slope triangle to the other (or: equal angles with horizontal)
• Equal slopes → parallel: lines with the same slope make the same angle with the horizontal, so they never intersect

Quick Check: Parallel or Not?

Line 1:
Line 2:

Are these lines parallel? Explain using the criterion.

Think before advancing to the answer...

Answer: Parallel Lines Have Different Intercepts

• Slope of Line 1:
• Slope of Line 2:
• Equal slopes → parallel ✓

Watch out: Same slope AND same intercept → the lines are identical, not parallel.
Parallel requires equal slopes and different y-intercepts.

Special Case: Vertical and Horizontal Lines

• Vertical lines: slope is undefined — all vertical lines are parallel to each other
• Horizontal lines: slope is 0 — all horizontal lines are parallel to each other
• The "equal slope" criterion covers these naturally: both vertical lines have "the same undefined slope" → parallel

Now: What About Perpendicular Lines?

You know perpendicular means meeting at a 90° angle.

• If a line has slope , what slope makes it perpendicular?
• The answer: — the negative reciprocal
• But why? Let's prove it with a rotation.

Rotation Proof: Slope After a 90° Turn

Rotating by 90° CCW about the origin gives .

The Product of Perpendicular Slopes Is Negative One

Original slope , perpendicular slope . Check the product:

Two non-vertical lines are perpendicular if and only if .

Quick Check: Find the Perpendicular Slope

A line has slope .

1. What slope is perpendicular to it?
2. Verify your answer using the product test.

Work it out before advancing...

Answer: Negative Reciprocal and Product Test

• Perpendicular slope:

Verify with the product:

Procedure: 4 written as → flip to → negate to

Second Proof: Pythagorean Theorem Approach

Two lines through the origin with slopes and .

Choose points and .

The lines are perpendicular if and only if at the origin .

Pythagorean Algebra Gives the Same Result

Compute the three distances:

Set and simplify:

Special Case: Horizontal and Vertical Lines

• A horizontal line () is always perpendicular to a vertical line (undefined slope)
• The product rule does not apply — you cannot compute
• This case is proven geometrically: they visibly meet at 90°

When to apply the product rule: only when both slopes are defined and nonzero.

Watch Out: "Negate" Is Not Enough

Common mistake: if slope is , thinking perpendicular slope is . We need the negative reciprocal — flip AND negate.

Verify: ✓ but ✗

Watch Out: Reciprocal Isn't Always Enough

Another trap: slopes and look like perpendicular partners.

They're reciprocals — but not negative reciprocals.

Draw the lines: both go upward to the right — they're not at 90°.

The perpendicular to slope is slope , not .

Key Takeaways

✓ Parallel criterion: equal slopes parallel (for distinct non-vertical lines)

✓ Perpendicular criterion: perpendicular (non-vertical, non-horizontal)

✓ Special cases: all vertical lines are parallel; horizontal ⊥ vertical

Perpendicular slope = negative reciprocal — flip the fraction AND negate

Slopes 2 and have product 1 — NOT perpendicular

Parallel lines: same slope, different y-intercepts

Coming Up in Lesson 2

Now you know why the criteria are true.

In Lesson 2, you will use them to:

• Write the equation of a line parallel to a given line through a point
• Write the equation of a line perpendicular to a given line through a point
• Find altitudes and perpendicular bisectors of geometric figures
• Verify right angles using the slope product test