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The diagram of a regular decagon ABCDEFGHIJ in which $\stackrel{¯}{AB}$and $\stackrel{¯}{CD}$ 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 $∠KBC$and $∠KCB$are the exterior angles of the regular decagon ABCDEFGHIJ.

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

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

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

Therefore, it can be obtained that:

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

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