Angles of Triangles

The Triangle Angle-Sum Theorem

If a triangle's angles measure $a^\circ$, $b^\circ$, and $c^\circ$, then $a + b + c = 180$. To find a missing angle, add the two you know and subtract from $180$.

The theorem also produces quick facts for special triangles. In a right triangle, the right angle uses $90^\circ$, so the two acute angles are complementary — they add to $90^\circ$. In an equilateral triangle, all three angles are equal, so each is $60^\circ$. In an isosceles triangle, the two base angles are congruent, so knowing the vertex angle gives you both: each base angle is half of what remains.

The Exterior Angle Theorem

Extend one side of a triangle past a vertex and you create an exterior angle — it sits outside the triangle and forms a linear pair with the interior angle at that vertex. The two interior angles that do not touch it are called its remote interior angles.

The Exterior Angle Theorem says the exterior angle equals the sum of the two remote interior angles. In the figure below, the exterior angle at $C$ measures $a + b^\circ$ — exactly the two far-away angles combined.

The reason is the angle sum itself. The interior angle at $C$ measures $180^\circ - a^\circ - b^\circ$, and the exterior angle is its supplement: $180^\circ - (180^\circ - a^\circ - b^\circ) = a^\circ + b^\circ$.

Setting up the equation

When the angles are algebraic expressions, both theorems become one-line equations. Three interior angles: add the expressions and set the sum equal to $180$. An exterior angle with its two remote interior angles: set the exterior expression equal to the sum of the other two.

The common trap is mixing the two setups — adding the exterior angle into a $180^\circ$ sum with its remote interior angles. The exterior angle is not inside the triangle; it only makes $180^\circ$ with the one interior angle it touches.

Worked examples

Try one yourself

Common questions