The Converse of the Theorem

Lesson 2 of 2: Reversing the Reasoning

In this lesson:

• State the converse and how it differs from the theorem
• Use the converse to test for a right angle

What You Will Be Able to Do

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

1. State the converse of the Pythagorean Theorem
2. Use the converse to decide whether a triangle is right
3. Tell when to apply the theorem versus the converse

Can Side Lengths Prove a Corner?

A carpenter measures a corner's three sides: 6, 8, 10. No protractor.

Can she conclude the corner is a perfect right angle?

Reversing a Statement Can Break It

"If it is a dog, then it is an animal." — True.

"If it is an animal, then it is a dog." — False.

Reversing a true statement does not keep it true.

Theorem and Converse, Side by Side

The arrows point opposite ways. These are two different claims.

Why This Converse Is True

If , build a right triangle with legs and .

Its hypotenuse is — same three sides, so SSS makes them congruent.

Test a Right Triangle Candidate

Sides 6, 8, 10. Is the longest side ?

By the converse, this is a right triangle.

Test a Triangle That Is Not Right

Sides 5, 7, 9. Longest side .

Since , it is not a right triangle.

Always Identify the Longest Side First

Step 1: Find the longest side — that is your candidate for .
Step 2: Test whether the other two squares sum to .

Test the wrong side and you get a false answer.

Your Turn: Is It a Right Triangle?

Sides 8, 15, 17. Longest side is 17.

Check whether . Decide before advancing.

Classify Any Triangle by Its Sides

Compare to : less means obtuse, equal means right, greater means acute.

Practice With Sides That Are Not Whole

Sides 1, , 2. Longest side is 2.

So this triangle is right.

Two Tools That Look Alike

You now have two tools using the same equation. A problem won't label which to use.

How do you tell what a problem is asking for?

Theorem Finds a Side; Converse Verifies

Theorem: you know it's right → find a side. Converse: you know the sides → check for a right angle.

Sort This Scenario Into a Tool

"A right triangle has legs 9 and 12. Find the hypotenuse."

You know it's right and want a side → this uses the theorem.

Your Turn: Sort These Scenarios

1. Garden corners measure 3 m, 4 m, diagonal 5 m. Are they square?
2. A 10-ft ladder's base is 6 ft out. How high does it reach?

Which uses the converse? Which uses the theorem?

Justify a Right Triangle From Scratch

Sides 20, 21, 29. Write a full justification that this is a right triangle.

Use complete converse language — no template.

Three Common Traps to Avoid Here

describes right triangles only — not every triangle
The theorem and converse are two facts, not one
Test against the longest side, or the answer flips

Two Directions of One Idea

✓ Theorem: right triangle → (find a side)
✓ Converse: → right triangle (verify an angle)

Next: 8.G.B.7 and 8.G.B.8 put both tools to work.