In Triangle Angle Sum and Exterior Angles you tore the corners off a paper triangle and saw that they make a straight line. That convinced you; now we prove it for every triangle at once, using the parallel-line theorems from Proving Theorems about Lines and Angles.
Why it works
Triangle Sum Theorem. Take any $\triangle ABC$. Through $B$, draw the line parallel to $\overline{AC}$ (there is exactly one). At $B$ three angles now sit side by side on that line: angle 1, then $\angle ABC$, then angle 3. Angle 1 and $\angle A$ are alternate interior angles, so they are congruent; so are angle 3 and $\angle C$. The three angles at $B$ make a straight angle, $180^\circ$, so $m\angle A + m\angle ABC + m\angle C = 180^\circ$.
Exterior Angle Theorem. Extend one side of a triangle. The angle outside, the exterior angle, forms a linear pair with the interior angle next to it. Both that pair and the three interior angles add to $180^\circ$, so the exterior angle equals the sum of the two remote interior angles (the two not touching it).
Key idea
The interior angles of a triangle add to $180^\circ$. An exterior angle equals the sum of the two remote interior angles.
Worked example: A missing angle
In $\triangle PQR$, $m\angle P = 47^\circ$ and $m\angle Q = 68^\circ$. Find $m\angle R$.
- $m\angle R = 180 - 47 - 68 = 65^\circ$.
Answer: $65^\circ$
Worked example: An exterior angle with algebra
Side $\overline{BC}$ of $\triangle ABC$ is extended to $D$. $m\angle ACD = (5x - 4)^\circ$, $m\angle A = (2x + 10)^\circ$ and $m\angle B = 40^\circ$. Find $m\angle ACD$.
- Exterior angle = sum of the remote interior angles: $5x - 4 = (2x + 10) + 40$.
- $5x - 4 = 2x + 50$, so $3x = 54$ and $x = 18$.
- $m\angle ACD = 5(18) - 4 = 86^\circ$. Check: $m\angle A = 46^\circ$, and $46 + 40 = 86$.
Answer: $m\angle ACD = 86^\circ$
Common mistake
Setting the exterior angle equal to the interior angle next to it, or to all three interior angles. The angle next to it is its supplement, and all three interior angles add to $180^\circ$. The exterior angle equals only the two remote angles. Check: exterior + adjacent interior $= 180^\circ$.