Why Slope Stays the Same on a Line

Lesson 1 of 2

In this lesson:

• Build slope triangles between points on a line
• Use similar triangles to prove slope is constant
• See why slope belongs to the line, not the points

Learning Objectives for This Deck

You should be able to:

1. Construct slope triangles between two points on a non-vertical line
2. Use similar triangles to prove the slope is the same between any two points

Will Any Two Points Really Agree?

Last lesson you picked two points and found the slope.

If you pick points near the origin and a friend picks points far out, are you guaranteed the same slope — or just lucky?

One Line, One Small Triangle

• Small triangle: rise , run , slope

Add a Bigger and an Offset Triangle

• Bigger: · Offset:

Coincidence, or Guaranteed Every Time?

A slope triangle has a horizontal run, a vertical rise, and the line as hypotenuse.

Both Triangles Have a Right Angle

• The run is horizontal, the rise is vertical — they meet at

They Share the Acute Angle Too

Both triangles share the angle where the line meets the horizontal.

• The line has one fixed direction
• So it makes the same angle with every horizontal run

AA: Two Angles Make Them Similar

By the AA criterion:

• Two pairs of equal angles the triangles are similar
• Both share the right angle and the acute angle

Similar Triangles Have Proportional Sides

Similar triangles have proportional corresponding sides:

Equal ratios — that is exactly "slope is constant."

The Argument as a Chain

1. Both triangles have a right angle
2. Both share the acute angle with the horizontal
3. By AA, the triangles are similar
4. Similar proportional sides equal rise/run

Quick Check: Verify the Numbers

Small triangle sides and ; large triangle sides and .

Does ?

Similar Is Not the Same as Congruent

• Different sizes, same ratio — that is all slope needs

One Exception: The Vertical Line

The argument needs a non-vertical line.

• A vertical line has run
• Slope is undefined

Fill In the Proof Yourself

Complete each blank:

• Both triangles have a ____ angle. (right)
• They share the angle where the ____ meets the horizontal. (line)
• By ____, the triangles are ____. (AA, similar)
• So ____ . (equals)

Watch Out: Two Common Errors

"The triangles must be congruent" — No. Similar is enough; equal ratios, not equal sizes.

"Slope depends on the points I pick" — No. Slope is a property of the line.

Your Turn: Draw and Justify

Here is a fresh line through and .

Draw two slope triangles of different sizes, compute both slopes, and write one sentence on why they must match.

Key Takeaways From This Lesson

✓ A slope triangle turns rise over run into a picture
✓ Two slope triangles on a line are similar (AA)
✓ Similar proportional sides slope is constant

Slope is a property of the line, not the points

Coming Up Next in Lesson Two

Now that slope is constant, you can pin a line down with a single equation. Next lesson uses one slope triangle — from the origin, then from the y-intercept — to derive and .