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

Short Answer

Expert verified

The measure of the fifth angle is 140°.

Step by step solution

01

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 n-2180.

02

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:

52180=3×180=540

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

03

Step 3. Description of Step.

Consider the measure of the fifth angle be x.

04

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.

40+80+115+165+x=540400+x=540x=540400x=140

Therefore, the value of x is 140.

05

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°.

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