Seemingly impossible question with the tangent-tangent theorem

Question

My teacher, after going over some notes with us, gave us this homework assignment and I found questions like the imgur link picture on it.

This question uses the tangent-tangent theorem, in which (major arc - minor arc) / 2 = angle X. The problem with this is that it requires 2 variables to solve any given problem (Or just one if it's an arc), but this has only one variable which is the angle. I can't do anything with just an angle. How am I supposed to solve this?

https://imgur.com/a/iE7HWpo

Answer 1

Let x = measure of the minor arc AD. Then the measure of major arc DPA is represented by 360 - x.

According to the tangent/tangent theorem: 84 = 1/2[(360 - x) - x]

360 - 2x = 168

2x = 192

x = 96 (measure of the minor arc)

So the major arc DPA = 360 - 96 = 264 (degrees)