Question

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

Short answer

The measure of the fifth angle is $140°$.

Step-by-step solution

Step 1. Write the theorem for the sum of interior angles for convex polygon.

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 $\left(n-2\right)180$.

Step 2. Find the value of the interior angle sum of the polygon.

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 $540°$.

Step 3. Description of Step.

Consider the measure of the fifth angle be x.

Step 4. Find the value of 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.

Step 5. Find the measure of the fifth angle.

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

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