The Converse (Classifying Triangles)
The converse of the Pythagorean Theorem lets you classify a triangle as right, acute, or obtuse — just from its three side lengths, no angles needed.
Introduction
The Pythagorean Theorem says: “right triangle → .” The converse says: “ → right triangle.” But what if the sides don't satisfy equality? The relationship between and tells you the triangle's type.
Past Knowledge
Pythagorean Theorem (8.1.1). Triples (8.1.2). Triangle classification by angles (5.1).
Today's Goal
Use side lengths to classify triangles as right, acute, or obtuse.
Future Success
Special right triangles (8.2), Law of Cosines (8.3.4).
Key Concepts
The Classification Test
Let be the longest side. Compare to :
| Comparison | Triangle Type | Why |
|---|---|---|
| Right | Pythagorean Theorem (converse) | |
| Acute | Largest angle < 90° | |
| Obtuse | Largest angle > 90° |
Theorem & Proof
Two-Column Proof: Converse of the Pythagorean Theorem
Given: with sides where
Prove:
Strategy: Construct a known right triangle with the same legs, then show the hypotenuses match → SSS Congruence.
| # | Statement | Reason |
|---|---|---|
| 1 | Given | |
| 2 | Construct right with legs and and right angle at | We can always construct a right triangle with any two legs |
| 3 | By Pythagorean Theorem on : | Pythagorean Theorem (8.1.1) — is right |
| 4 | Substitute from step 1 | |
| 5 | SSS Congruence (sides match) | |
| 6 | CPCTC — corresponding angles of congruent triangles are equal |
∎ If the sides satisfy , the triangle must contain a right angle.
Worked Examples
Right Triangle Check
Sides: 9, 40, 41. Classify the triangle.
✓
Right triangle (9-40-41 is a Pythagorean triple!)
Acute Triangle
Sides: 5, 6, 7. Classify.
Longest side: 7. Compare:
→
Acute triangle
Obtuse Triangle
Sides: 4, 5, 8. Classify.
Longest side: 8. Compare:
→
Obtuse triangle
Common Pitfalls
Not Using the Longest Side as c
Always put the longest side on the right side of the comparison. If you use a shorter side as , you'll get the wrong classification.
Confusing > and <
If , the angle is smaller than 90° → acute. If , the angle is bigger → obtuse. Think: bigger means bigger angle opposite it.
Real-Life Applications
Carpentry — Checking Square Corners
A carpenter who measures 3 feet and 4 feet along two edges of a frame knows the diagonal should be exactly 5 feet. If it's less than 5, the angle is obtuse (frame is “too open”). More than 5 means acute (“too closed”).
Surveying — Land Shape Analysis
Surveyors measure three distances between landmarks. The converse lets them determine the type of triangle formed, which affects how they compute the enclosed area.
Practice Quiz
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