Exterior angle property

When you extend a side of a triangle, the angle formed outside is called an exterior angle. There is a beautiful rule about it.

Idea

Take triangle $ABC$. Extend side $BC$ beyond $C$ to a point $D$. The angle $\angle ACD$ (between the extension and the slanting side $CA$) is the exterior angle at $C$.

Exterior angle property. This exterior angle equals the sum of the two interior angles at the other two vertices ($A$ and $B$ , called opposite interior angles): $\angle ACD = \angle A + \angle B.$

Proof. The interior angle at $C$ and its exterior angle $\angle ACD$ form a linear pair, so $\angle ACB + \angle ACD = 180^\circ.$ By the angle sum property, $\angle ACB + \angle A + \angle B = 180^\circ.$ Subtracting, $\angle ACD = \angle A + \angle B$. ✓

Why useful. Often the exterior angle is given, and you need to find an interior angle, or vice versa. Without this rule, you would have to first compute the interior angle at $C$, then use the angle sum to get the others , two steps. With the rule, one step.

A corollary: an exterior angle is always greater than each of the two opposite interior angles (because their sum equals it, and each is non-negative).

Worked examples

Example 1. In triangle $ABC$, $\angle A = 50^\circ$, $\angle B = 70^\circ$. Find the exterior angle at $C$.

$\angle$ exterior at $C = \angle A + \angle B = 50 + 70 = 120^\circ$.

Example 2. An exterior angle of a triangle is $130^\circ$. The two opposite interior angles are in ratio $2:3$. Find them.

Sum is $130^\circ$. Ratio $2:3$ means $\tfrac{2}{5}(130) = 52^\circ$ and $\tfrac{3}{5}(130) = 78^\circ$.

Example 3. In triangle $ABC$, the exterior angle at $C$ is $115^\circ$ and $\angle A = 45^\circ$. Find $\angle B$.

$\angle A + \angle B = 115 \Rightarrow \angle B = 115 - 45 = 70^\circ$.

Example 4. Verify: in an equilateral triangle, each exterior angle is $120^\circ$.

Each interior is $60^\circ$. Exterior $=$ sum of other two $= 60 + 60 = 120^\circ$. ✓ (Or: $180 - 60 = 120$.)

Try it yourself

1. In triangle $ABC$, $\angle A = 40^\circ, \angle B = 75^\circ$. Find exterior at $C$.
2. Exterior angle at $A$ is $110^\circ$; $\angle B = 50^\circ$. Find $\angle C$.
3. In a right triangle, find the exterior angle at the right-angle vertex.
4. The three exterior angles (one at each vertex) of a triangle sum to what?
5. In triangle $ABC$, exterior at $C = 4x$ and the two opposite interiors are $x+10$ and $2x-5$. Find $x$.
6. Why is an exterior angle always greater than either opposite interior?
7. The exterior angle of an isosceles triangle (at the apex) is $130^\circ$. Find each base angle.
8. Can the exterior angle of a triangle be $50^\circ$? Justify with the property.

Activity

Draw any triangle. Extend each side outward. Measure each exterior angle. Verify the rule: exterior $=$ sum of two opposite interiors. Also verify the three exterior angles sum to $360^\circ$.