PROVE — Concept Note

Concept

For positive side lengths a, b and c of a triangle, $a^2$ + $b^2$ = $c^2$ implies the angle opposite c is a right angle. To prove this, compare the given triangle with a constructed right triangle.

Key Ideas

• In the forward theorem c is the hypotenuse.
• Because a and b are positive, $c^2$ = $a^2$ + $b^2$ is greater than both $a^2$ and $b^2$. Positive lengths therefore make c the longest side.
• A constructed right triangle and SSS congruence establish the original angle; do not assume it in the diagram.

Rule / Property

Construct perpendicular legs a and b. Use the forward theorem to find their hypotenuse. Match all three sides, then match the included angle by SSS.

Visual Recall

Mini Example

Given: BC = a, CA = b, AB = c and $a^2$ + $b^2$ = $c^2$. Prove ∠C = 90°.

Verify numerically: $8^2$ + ${15}^2$ = 64 + 225 = 289 = ${17}^2$.

Common Mistake

Using the longest side as a leg, or marking the original angle right before proving it.

Quick Recall

Do sides 9, 12, 15 form a right triangle? Justify the angle location.