Triangle identities · trigonometric identity
A + B + C = 180° — Angles of a triangle add to 180°
When to use it
Two angles are known and the third must be found before any side work starts — this is the step that turns 'two angles' into a usable pair for the sine rule. It also tells you when a proposed triangle cannot exist.
Why it is true
Draw the line through the top vertex parallel to the opposite side. It makes alternate angles equal to the two base angles, and the three angles along one straight line add to 180°, so A + B + C = 180°.
The full line-by-line version is on the proof page for angles of a triangle add to 180°; the “how would I find this myself” version is in the derivation.
Where it comes from
use it as an equation for the missing angle. In practice nobody proves this identity at the whiteboard; they solve it. Treat it as a one-step equation and watch the unit trap.
- C = π − A − BRearrange for the unknown angle. · Angles of a triangle add to 180°
- A = π/3, B = π/4 → C = 5π/12π − π/3 − π/4 = 12π/12 − 4π/12 − 3π/12 = 5π/12 (75°).
- A = 100°, B = 90° → no triangleThe sum already exceeds 180° before C is chosen — the identity is also an existence test.
Worked examples
- A = 30°, a = 6, b = 10 — solve the triangleTwo triangles: B ≈ 56.44° with C ≈ 93.56° and c ≈ 11.98, or B ≈ 123.56° with C ≈ 26.44° and c ≈ 5.34.
Checked on these triangles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| 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 |
| 45° | 3.141593 | 3.141593 | yes |
| 53° | 3.141593 | 3.141593 | yes |
| 60° | 3.141593 | 3.141593 | yes |
| 67.5° | 3.141593 | 3.141593 | yes |
| 75° | 3.141593 | 3.141593 | yes |
| 90° | 3.141593 | 3.141593 | yes |
| 120° | 3.141593 | 3.141593 | yes |
| 135° | 3.141593 | 3.141593 | yes |
| 150° | 3.141593 | 3.141593 | yes |
The letters are not free
a, b, c and A, B, C here are one triangle's sides and angles: A + B + C = 180° and the sides follow the sine rule. Feed the same six letters with random values and the equation fails — that is a property of the formula, not a bug in the tool.
The mistake students make
Subtracting from 360° instead of 180°, or mixing degrees and radians in the same line. A triangle's angle sum is a half turn — π radians, not 2π.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for triangle identities.
Related identities
- a/sin A = b/sin BLaw of sinesdetailsproof
- a² = b² + c² − 2bc·cos ALaw of cosinesdetailsproof
- sin θ = cos(90° − θ)Cofunction identity for sinedetailsproof
Category hub: Triangle identities · all identities: /identities