Question

$\overline{BC}$is one side of a regular n-gon. The sides next to $\overline{BC}$ are extended to meet at W. Find the measure of $\angle W$ in terms of n.

Short answer

The measure of $\angle W$in terms of $n$is$\frac{180°\left(n-4\right)}{n}$

Step-by-step solution

Step 1. Draw the diagram of a regular n-gon in which sides next to BC are extended to meet at w.

The diagram of a regular n-gon in which sides next to $\overline{BC}$ are extended to meet at W is:

Step 2. Description of step.

The measure of each exterior angle of a regular polygon with n sides is given by:

$\frac{360°}{n}$

Step 3. Description of step.

From the diagram it can be noticed that the angles $\angle WBC$and $\angle WCB$are the exterior angles of the regular n-gon.

Therefore, the measures of the angles $\angle WBC$and $\angle WCB$is $\frac{360°}{n}$.

The sum of the angles of the triangle is $180°$.

Therefore, the sum of the angles of the triangle $\Delta WBC$is $180°$.

Therefore, it can be obtained that:

$$\begin{gathered} \angle WBC+\angle WCB+\angle BWC=180° \\ \frac{360°}{n}+\frac{360°}{n}+\angle BWC=180° \\ \frac{720°}{n}+\angle BWC=180° \\ \angle BWC=180°-\frac{720°}{n} \\ \angle BWC=\frac{180°n-720°}{n} \end{gathered}$$

Therefore, the measure of the angle $\angle W$in terms of is $\frac{180°\left(n-4\right)}{n}$.