Hello Jacob,

Suppose, as in your problem, that one line is tangent to a given circle at point A, and a second line is tangent to this circle at point C. Suppose further that the lines intersect at the point B (so we name the lines AB and CB.) Let **θ be the measure of ∠CBA (in degrees.)** We are given that **θ = 60º**. Let us also use degree measure to measure arcs. (That is, the measure of a given arc on a circle will be the same as the measure of the central angle that cuts off this arc.) Note that the points A and C cut off two arcs on the circle. The smaller is (called arc AC) has measure less than 180º. The larger arc (major arc) is called arc APC, and it has measure greater than 180º. **The following key fact gives the relationship between the measure of ∠CBA and the measure of arc AC is**

**Measure(Arc AC) = 180º - θ**

Given that θ = 60°, and Measure(Arc AC) = 4x, we have

4x = 180º - 60º

4x = 120º

**x = 30º**

Hope that helps! Let me know if you need any further explanation.

William

PS. Note that you can also find x using the information about the major arc. Since the measure of arc AC is 180º - θ, then Measure(Arc APC) = 360º- (180º- θ) = 180º + θ = 240º. Set this equal to 9x - 30 and solve for x, and you will obtain the same result as above.