In a circle centered on O, the angle between the diameter MN and the chord NK is 67. Find the angle KMN and MOK.

Since MN is the diameter of the circle, the degree measure of arc MKN and arc MN are 180. Then the inscribed angle MKN is 180/2 = 90, and therefore triangle MKN is right-angled.

Then in a right-angled triangle MKN the angle KMN = (180 – MKN – MNK) = (180 – 90 – 67) = 23.

The MOC triangle is isosceles, since OK = OM = R, then the angle KMO = MKO = 23.

Then the angle MOK = (180 – KMO – MKO) = (180 – 23 – 23) = 134.

Answer: The MOK angle is 134, the KMN angle is 23.