Theorem

Angle sum of a triangle

Geometry • Triangle • Angles of a Triangle

Formula

∠ A + ∠ B + ∠ C = 18 0^{\circ}

In any plane triangle the three interior angles sum to

18 0^{\circ}

; equivalently, an exterior angle equals the sum of the two interior angles not adjacent to it.

How to use it

1Given two angles of a triangle, subtract from $18 0^{\circ}$ for the third.
2With the angles given as a ratio, write them as $2 k$ , $3 k$ , $4 k$ and solve $9 k = 18 0^{\circ}$ .
3Apply it to a sub-triangle cut off by a cevian to reach an angle at an interior point.

In triangle

P QR

∠ P = 5 2^{\circ}

and

∠ Q = 7 1^{\circ}

. Find

∠ R

and the exterior angle at

R

Step-by-step

1
$∠ R = 18 0^{\circ} - 5 2^{\circ} - 7 1^{\circ} = 5 7^{\circ}$
Why: The three interior angles close to $18 0^{\circ}$ .
2
$18 0^{\circ} - 5 7^{\circ} = 12 3^{\circ} = 5 2^{\circ} + 7 1^{\circ}$
Why: The exterior angle is the supplement of $∠ R$ , and equals the two remote interior angles added.

Answer

∠ R = 5 7^{\circ}

and the exterior angle at

R

12 3^{\circ}

Watch out

It is the INTERIOR angles that sum to

18 0^{\circ}

, and only for a plane triangle; a quadrilateral's interior angles sum to

36 0^{\circ}

, and an

n

-gon's to

(n - 2) 18 0^{\circ}

Also known as

angle sum property, angles of a triangle add to 180, sum of interior angles of a triangle, exterior angle theorem, third angle of a triangle