With this information we will find the range of possible values of x. Since x is an angle formed by a chord and a tangent, we will begin by recalling the Tangent and Intersected Chord Theorem (Theorem 10.14).
Tangent and Intersected Chord Theorem
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of its intercepted arc.
The intercepted arc of x is AB.
Therefore, by using the theorem we can write x in term of mAB.
x=1/2mAB
Now we should determine the interval of mAB. By the first piece of the given information, we can conclude that A and B are any two points on the circle.
From here, we can conclude that the values of mAB are between 0^(∘) and 360^(∘). Moreover, since AB is not a diameter, mAB cannot be 180^(∘).
0 < mAB < 360, mAB≠ 180
Next, we will multiply each side of the above statement by 12 and substitute x for 12mAB.
0 < 1/2mAB < 180, 1/2mAB≠ 90
⇕
0 < x < 180, x≠ 90
Therefore, the values of x are between 0^(∘) and 180^(∘) but x cannot be 90^(∘).
