Question

the diagram shows a convex polygon. what is the value of z? z = □° (diagram includes angles: ( z ), ( z - 8^circ ), ( z - 31^circ ), ( z + 13^circ ), ( 98^circ ))

Explanation:

Step1: Recall the sum of exterior angles of a convex polygon.

The sum of the exterior angles of any convex polygon is \(360^\circ\). So we can set up an equation by adding all the given exterior angles and setting it equal to \(360^\circ\).
The angles are \(z\), \(z - 8^\circ\), \(98^\circ\), \(z + 13^\circ\), and \(z - 31^\circ\). So the equation is:
\[z+(z - 8)+98+(z + 13)+(z - 31)=360\]

Step2: Simplify the left - hand side of the equation.

First, combine like terms:
\[

$$\begin{align*} z+z - 8+98+z + 13+z - 31&=(z + z+z + z)+(- 8 + 98+13 - 31)\\ &=4z+(90 + 13-31)\\ &=4z+(103 - 31)\\ &=4z + 72 \end{align*}$$

\]
So our equation becomes \(4z+72 = 360\).

Step3: Solve for \(z\).

Subtract 72 from both sides of the equation:
\[4z+72-72=360 - 72\]
\[4z=288\]
Then divide both sides by 4:
\[z=\frac{288}{4}=72\]

Answer:

\(72\)