Find ∠KOM if the degree measures of the arcs KO and OM are 112 ° and 170 °, respectively.

Points K, O, and M lie on a circle. Their order can be different.
Suppose that K and M lie on opposite sides of the point O, then:
112 ° + 170 ° = 282 °,
360 ° -282 ° = 78 ° is the degree measure of the arc on which ∠KOM rests.
∠KOM is an inscribed angle, which means that it is equal to half the degree measure of the arc on which it rests:
∠KOM = 78 °: 2 = 39 °.
Suppose that K and M lie on the same side of the point O, then:
170 ° -112 ° = 58 ° -degree measure of the arc on which ∠KOM rests.
∠KOM = 58 °: 2 = 29 °.
Answer: 39 ° or 29 °.