$ABCDEFGHIJ$is a regular decagon. If sides $\overline{AB}$and $\overline{CD}$ extended to meet at K, find the measure of $\angle K$.

The diagram of a regular decagon ABCDEFGHIJ in which $\overline{AB}$and $\overline{CD}$ extended to meet at Kis:

The measure of each exterior angle of a regular polygon with n sides is given by: $\frac{360°}{n}$

The decagon has 10 sides.

Therefore, $n=10$.

The measure of each exterior angle of a regular polygon with 10 sides is given by: $\frac{360°}{10}=36°$

From the diagram it can be noticed that the angles $\angle KBC$and $\angle KCB$are the exterior angles of the regular decagon ABCDEFGHIJ.

Therefore, the measures of the angles $\angle KBC$and $\angle KCB$is $36°$.

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

Therefore, the sum of the angles of the triangle$\angle KBC$is $180°$.

Therefore, it can be obtained that:

$$\begin{gathered} \angle KBC+\angle KCB+\angle BKC=180° \\ 36°+36°+\angle BKC=180° \\ 72°+\angle BKC=180° \\ \angle BKC=180°-72° \\ \angle BKC=108° \end{gathered}$$

Therefore, the measure of the angle $\angle K$is $180°$.