Chapter 12.3, Problem 5C

To determine

To find : The measure of the fifth exterior angle of a convex pentagon.

Answer to Problem 5C

The measure of the fifth exterior angle of the given pentagon is ${173}^{\circ }$ .

Explanation of Solution

Given information : Here, the polygon is convex pentagon. Its four exterior angles are ${14}^{\circ }$ , ${87}^{\circ }$ , ${56}^{\circ }$ and ${30}^{\circ }$ .

Formula used :

The sum of the measure of the all interior angles of a convex n-gon is $\left(n-2\right)\times {180}^{\circ }$

Measure of exterior angle $={180}^{\circ }-$ interior angle

Calculation :

Here, polygon is convex pentagon i.e. $n=5$

Let, the measures of the interior angles of pentagon are

  $m\angle 1,m\angle 2,m\angle 3,m\angle 4,m\angle 5$

The sum of the measure of the all interior angles of a convex pentagon is

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

i.e.

  $\begin{array}{l}(m\angle 1+m\angle 2+m\angle 3+m\angle 4+m\angle 5)\\ ={540}^{\circ }\end{array}$

Therefore, the sum of the measures of the exterior angles of the given convex pentagon is

  $\begin{array}{l}\left({180}^{\circ }-m\angle 1\right)+\left({180}^{\circ }-m\angle 2\right)+\left({180}^{\circ }-m\angle 3\right)+\left({180}^{\circ }-m\angle 4\right)+\left({180}^{\circ }-m\angle 5\right)\\ =\left({180}^{\circ }+{180}^{\circ }+{180}^{\circ }+{180}^{\circ }+{180}^{\circ }\right)-(m\angle 1+m\angle 2+m\angle 3+m\angle 4+m\angle 5)\\ =\left(5\times {180}^{\circ }\right)-\left({540}^{\circ }\right)\\ ={900}^{\circ }-{540}^{\circ }\\ ={360}^{\circ }\end{array}$

Sum of the given four exterior angles is

  $\begin{array}{l}{14}^{\circ }+{87}^{\circ }+{56}^{\circ }+{30}^{\circ }\\ ={187}^{\circ }\end{array}$

Therefore, the required fifth angle is

  $\begin{array}{l}{360}^{\circ }-{187}^{\circ }\\ ={173}^{\circ }\end{array}$