Understanding the Context

A right-angled triangle satisfies the Pythagorean theorem:
a² + b² = c², where c is the hypotenuse (the side opposite the right angle) and a, b are the other two sides. This property makes identifying a right triangle relatable and crucial in many practical and theoretical applications.

Step-by-Step Method to Verify a Right-Angled Triangle

1. Measure All Three Sides

Key Insights

Begin by accurately measuring the lengths of all three sides of the triangle using a ruler, measuring tape, or other reliable tools—if physical. In digital contexts or theoretical problems, referenced side lengths from a diagram or equation may be used.

2. Apply the Pythagorean Theorem

Arrange the side lengths such that the largest side is identified as the hypotenuse (opposite the largest angle). Then test the equation:
a² + b² = c²
If this equality holds true, the triangle is right-angled.

Example:
Sides: 3 cm, 4 cm, 5 cm
Check: 3² + 4² = 9 + 16 = 25; and 5² = 25
Thus, 3-4-5 triangle is right-angled.

3. Use Angle Measurement (for Physical Triangles)

Final Thoughts

If drawing or measuring sides, use a protractor to directly measure all three internal angles. A triangle is right-angled if any angle measures exactly 90 degrees. However, due to precision limits in manual protractor use, relying on the Pythagorean theorem avoids measurement error in many situations.

4. Use Trigonometric Ratios (Alternative Verification)

For triangles with known side ratios, use trigonometric functions:

• If sin(θ) = opposite/hypotenuse = 1, then ∟θ exists.
  
• Similarly, cosθ = adjacent/hypotenuse = 0 implies a right angle.

5. Confirm with Area-Based Methods (Less Common)

While not direct, comparing area formulas (e.g., base × height / 2) with Heron’s formula can provide consistency—but only after confirming right-angle geometry via Pythagoras first.