Why does a triangle have an interior angle of 180 degrees?

Question

Every other regular polygon has 360, why is a triangle different?

Answer 1

Suppose you have triangle ABC. Draw a line, $l$, parallel to AB going through C. You will then see that the acute angle formed by $l$, A, and C will be equal to angle A, since they are alternate interior angles in parallel lines. The same can be said about the angle formed by $l$, B and C, which will be equal to angle B. Therefore, A+B+C must equal $180$ degrees.

Answer 2

The dotted line through $C$ is parallel to $AB$.

Answer 3

As @dxiv mentioned, the sum of exterior angles $\alpha_i$ of any convex polygon is $360^\circ$:

$$E = \sum_i \alpha_i = 360^\circ$$

You can justify this to yourself through a visual argument, as given here.

The complement of any exterior angle $\alpha_i$ is an interior angle of value $180^\circ - \alpha_i$. Since a triangle has three such exterior angles, the sum of its interior angles must be:

$$I = \sum_{i=1}^3 (180^\circ - \alpha_i)$$

Which can be rewritten:

$$I = 3\cdot 180^\circ - \sum_{i=1}^3 \alpha_i$$

Recalling the sum of exterior angles $E = \sum \alpha_i = 360^\circ$,

$$ \begin{align} I &= 3\cdot 180^\circ - 360^\circ \\ &= 180^\circ \end{align} $$

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You can use this same argument to derive the general formula for convex polygons of $n$ sides:

$$ \begin{align} I &= n\cdot 180^\circ - \sum_{i=1}^n \alpha_i \\ &= (n - 2) \cdot 180^\circ \end{align} $$