Angles of Triangles

The interior angle sum

The three angles inside any triangle add to $180^\circ$. If a triangle has angles of $38^\circ$ and $84^\circ$, the third angle is $180 - 38 - 84 = 58^\circ$. Add the two angles you know, then subtract from $180$.

Special triangles make this even faster. A right triangle spends $90^\circ$ on its right angle, so the two acute angles add to $90^\circ$. An equilateral triangle splits $180^\circ$ evenly, so each angle is $60^\circ$. In an isosceles triangle, the two angles opposite the equal sides match.

Exterior angles

Extend one side of a triangle past a vertex and you create an exterior angle. The exterior angle and the interior angle beside it form a linear pair, so they add to $180^\circ$.

Here is the shortcut: the exterior angle equals the sum of the two remote interior angles — the two angles of the triangle farthest from it. In the figure, one side is extended past a vertex, and the exterior angle marked $1 + 2$ equals the sum of the two remote interior angles $1$ and $2$. It works because the interior angle beside the exterior angle takes $180^\circ$ minus the other two angles, which leaves the exterior angle equal to exactly those two angles combined.

When the angles are expressions

Some problems label angles with expressions like $2x + 10^\circ$ instead of numbers. The plan does not change: write what the angles must add to — $180^\circ$ for the three interior angles of a triangle — then solve the equation for $x$. If the question asks for an angle measure rather than $x$, remember to substitute $x$ back into the expression at the end.

Worked examples

Try one yourself

Common questions