Abcdefghij is a regular decagon if sides CD and ab are extended to form angle k what is the measure of angle k?

Answer 1

The total degree measure in a decagon is 180(8) since a decagon can be broken up into 8 triangles. In a regular decagon, each angle has the same measure: 180(8)/10=18(8)=144. The supplementary angle, 36, is therefore the angle between the side bc and (each of) the two extended sides ab, CD outside of the decagon. The remaining angle, k, of the triangle thus formed is 180-2(36)=180-72=108.

A sneaky way to get the same answer is to notice that if we extend every other side of the regular decagon, we get a (larger) regular pentagon. The angle k is one of these angles, so it is 108.