Question

What is the measure (in degrees) of the smallest interior angle of a triangle in which the exterior angle measures have the ratio $3:4:5$?

Answer 1

30 degrees

Explanation:

Since the exterior angle measures have the ratio $3:4:5$, they are $3x:4x:5x$ for some value of $x$. Each exterior angle is supplementary to an interior angle, so the measures of the interior angles of the triangles are $180^\circ - 3x$, $180^\circ - 4x$, and $180^\circ - 5x$. The sum of the interior angles of a triangle equals $180^\circ$, so we have

\[(180^\circ -3x) + (180^\circ - 4x) + (180^\circ - 5x) = 180^\circ.\]Simplifying this equation gives $-12x = -360^\circ$, so $x = 30^\circ$. Therefore, the smallest interior angle has measure $180^\circ - 5x = \boxed{30^\circ}$.

Answer 2

Let

3x----> angle exterior 1

4x----> angle exterior 2

5x----> angle exterior 3

we know that

The exterior angles of any polygon add up to 360 degrees

so

3x+4x+5x=360-------> 12x=360----------> x=30°

The largest of exterior angles is equal to the smallest of interior angles

so

the largest of exterior angles is 5x------> 5*30=150°

the smallest of interior angles is 180°-150°=30°

therefore

the answer is

the smallest of interior angle measure 30 degrees