Pythagoras Theorem
102. The Converse Connection · When sides match, the angle snaps to 90°
Whenever a² + b² = c², the angle opposite side c is exactly 90°.
The Converse of Pythagoras says: if a² + b² = c² for the three sides of a triangle, then the triangle is right-angled with the right angle opposite to side c. This lets us check rightness from side lengths alone.
What this lesson covers
Try to break it
Try breaking the equation a² + b² = c². Notice the angle at C stops being a right angle. The rule only holds when the sides match perfectly.
How you build it
Construct a right triangle with the right angle at C — its sides will satisfy a² + b² = c².
- Mark point C — the right-angle corner.
- Draw a line through C — this is the first leg.
- Construct a line perpendicular to it at C — the second leg.
- Mark point A on the first line.
- Mark point B on the perpendicular line.
- Draw the hypotenuse AB to complete the right triangle.
The proof, step by step
Prove that if a² + b² = c², the angle opposite side c is 90°.
- Given: In ΔABC, AB² = AC² + BC².
- Construct ΔA'B'C' such that ∠C' = 90°, A'C' = AC, B'C' = BC.
- By Pythagoras theorem in ΔA'B'C', A'B'² = A'C'² + B'C'².
- Substituting given values, A'B'² = AC² + BC² = AB².
- Therefore, A'B' = AB.
- By SSS congruence, ΔABC ≅ ΔA'B'C'.
- Hence, ∠C = ∠C' = 90°.
Worked example
In ΔPQR, PQ = 13 cm, QR = 12 cm, and RP = 5 cm. Which of the following is true?
Here, RP² + QR² = 5² + 12² = 25 + 144 = 169 = 13² = PQ². Since the square of the longest side equals the sum of the squares of the other two sides, by the converse of the Pythagorean theorem, the angle opposite PQ (which is ∠R) is 90°.
- ∠P = 90°
- ∠Q = 90°
- ∠R = 90° — correct
- None of these