Four of the angles of a pentagon have measures 40, 80, 115, and 165. Find the measure of the fifth angle.

The theorem for the sum of interior angles for convex polygon states that the sum of measures of the interior angles of a convex polygon with sides is $\left(n-2\right)180$.

By using the theorem for the sum of interior angles for convex polygon, the sum of the interior angles for the polygon having 5 sides is given by:

$\begin{array}{c}\left(5-2\right)180=3\times 180\\ =540\end{array}$

Therefore, the sum of interior angles for the pentagon is $540°$.

Consider the measure of the fifth angle be x.

The sum of interior angles for the pentagon is $540°$, the measures of the four angles are 40, 80, 115, 165 and the measures of the fifth angle is x.

Therefore, the equation for the given problem is:

$40+80+115+165+x=540$

Now solve the equation to find the value of x.

$$\begin{gathered} 40+80+115+165+x=540 \\ 400+x=540 \\ x=540-400 \\ x=140 \end{gathered}$$

Therefore, the value of x is 140.

The value of x is 140 and the measures of the fifth angle is x.

Therefore, the measures of the fifth angle is $140°$.