Triangle identities · proof ·what it is and when to use it
Proof: A + B + C = 180°
Angles of a triangle add to 180° — proved by parallel line through the apex. Every line below says which rule it uses, so nothing has to be taken on faith.
The proof
Euclid's argument. It is worth reading once because it shows where the statement comes from: the parallel postulate, which is also why the angle sum of a triangle on a sphere is more than 180°.
- draw ℓ through C parallel to ABThe construction Euclid uses (Elements I.32).
- the angle between ℓ and CA equals A; the angle between ℓ and CB equals BAlternate angles made by a transversal across parallel lines.
- A + B + C = 180°Those three angles lie on one side of the straight line ℓ.
The sampler checks this claim in radians: A + B + C against π, on triangles built from three corner points.
Where the proof stops applying
The variables are one triangle's angles and sides, so they obey A + B + C = 180° and the sine rule. Drop that constraint and the equation is simply not true — which is why the sampler here generates triangles rather than random numbers.
The same claim, checked numerically
A proof is not the same thing as a check, and this page shows both: the reasoning above, and the first few triangles fed to the same engine behind the verifier. If they ever disagreed, the data would be broken — the build would fail before publishing (accuracy policy).
| Triangle (A) | Left side | Right side | Agree |
|---|---|---|---|
| 7.5° | 3.141593 | 3.141593 | yes |
| 15° | 3.141593 | 3.141593 | yes |
| 18° | 3.141593 | 3.141593 | yes |
| 22.5° | 3.141593 | 3.141593 | yes |
| 30° | 3.141593 | 3.141593 | yes |
| 37° | 3.141593 | 3.141593 | yes |
Related
- How would I find A + B + C = 180° myself? — the derivation, which is a different question from the proof.
- Angles of a triangle add to 180°: when to use it — the practical side.
- All triangle identities · proof index