Reverse the theorem
Converse of the Pythagorean Theorem
The converse tells you whether a triangle is a right triangle, without measuring angles. Learn the definition, proof, and how to classify any triangle as acute, right, or obtuse.
Definition
What Is the Converse of the Pythagorean Theorem?
The Pythagorean theorem states: If a triangle is a right triangle, then
a² + b² = c².
The converse reverses this statement: If a² + b² = c² for the three sides of a
triangle, then the triangle is a right triangle.
| Statement | Direction | |
|---|---|---|
| Pythagorean Theorem | Right triangle → a² + b² = c² |
Forward |
| Converse | a² + b² = c² → Right triangle |
Backward |
The converse is not automatically true for every theorem, it must be proven separately. For the Pythagorean theorem, the converse is true and can be established with SSS congruence. That means side lengths alone are enough to determine whether a triangle contains a right angle.
Classification
Classifying Triangles with the Converse
The converse can be extended to classify any triangle, not just right triangles. Let
c be the longest side. Compare a² + b² to c².
Right Triangle
a² + b² = c²
The angle opposite c is exactly 90°.
Acute Triangle
a² + b² > c²
All angles are less than 90°.
Obtuse Triangle
a² + b² < c²
The angle opposite c is greater than 90°.
Memory tip: Think of c² as a target.
Hit the target exactly (=) → Right triangle
Overshoot the target (>) → Acute triangle
Undershoot the target (<) → Obtuse triangle
Proof
Proof of the Converse
This page uses a short SSS-congruence proof rather than the full Euclidean treatment. It is enough to justify why the converse is valid.
| Step | Reasoning |
|---|---|
| Step 1 | Construct a second triangle with legs a and b and a right angle between them. Call its hypotenuse x. |
| Step 2 | By the Pythagorean theorem, x² = a² + b². |
| Step 3 | We are given a² + b² = c², so x² = c², which means x = c. |
| Step 4 | Both triangles now have sides a, b, and c. By SSS congruence, they are identical. |
| Step 5 | Since the constructed triangle has a right angle, the original triangle must also have a right angle. ∎ |
For four complete proofs of the Pythagorean theorem itself, see Pythagorean Theorem Proof.
Worked examples
Worked Examples
These four examples all answer the same question: from the side lengths alone, is the triangle acute, right, or obtuse?
Is 3-4-5 a right triangle?
| Identify longest side | c = 5 |
|---|---|
Calculate a² + b² |
3² + 4² = 9 + 16 = 25 |
Calculate c² |
5² = 25 |
| Compare | 25 = 25 ✓ |
a² + b² = c²)Is 5-7-9 a right triangle?
| Identify longest side | c = 9 |
|---|---|
Calculate a² + b² |
5² + 7² = 25 + 49 = 74 |
Calculate c² |
9² = 81 |
| Compare | 74 < 81 |
a² + b² < c²)Is 4-5-6 a right triangle?
| Identify longest side | c = 6 |
|---|---|
Calculate a² + b² |
4² + 5² = 16 + 25 = 41 |
Calculate c² |
6² = 36 |
| Compare | 41 > 36 |
a² + b² > c²)Is 8-15-17 a right triangle?
| Identify longest side | c = 17 |
|---|---|
Calculate a² + b² |
8² + 15² = 64 + 225 = 289 |
Calculate c² |
17² = 289 |
| Compare | 289 = 289 ✓ |
a² + b² = c²)Calculator
Verify Your Triangle Instantly
Use the calculator below to check whether any set of three side lengths forms a right triangle.
Triangle Verification Calculator
Enter three side lengths and instantly see whether the triangle is right, acute, or obtuse.
Verify Triangle →FAQ
Frequently Asked Questions
If you measure the three sides of a triangle and find that a² + b² = c², where
c is the longest side, then the triangle must have a right angle opposite that
side. You do not need to measure any angles.
Yes. Unlike some converses in mathematics that fail, this one is always true. A standard
proof uses SSS congruence to show that a triangle satisfying a² + b² = c² must
match a right triangle with the same three side lengths.
Let c be the longest side. Compare a² + b² to c². If
they are equal, the triangle is right. If a² + b² > c², it is acute. If
a² + b² < c², it is obtuse.
The theorem goes forward: knowing the triangle is right, you can relate its sides with
a² + b² = c². The converse goes backward: knowing the side lengths, you can
decide whether the triangle is right.
Yes. The 3-4-5 rule used in construction is a direct application of the converse. If a corner measures 3 units on one side, 4 on the other, and 5 across the diagonal, the corner is square.
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