Answer: [tex]59^{\circ}[/tex]
Step-by-step explanation:
In triangle ABC, we know that [tex]\angle ABC=130^{\circ}-\theta[/tex]. So, by the inscribed angle theorem, minor arc AC is equal to [tex]260^{\circ}-2\theta[/tex].
Thus, major arc AC is equal to [tex]100^{\circ}+2\theta[/tex].
Using the fact that the angle formed by the two tangents is 38 degrees,
[tex]\frac{(100^{\circ}+2\theta)-(260^{\circ}-2\theta)}{2}=38^{\circ}\\\\100^{\circ}+2\theta-260^{\circ}+2\theta=76^{\circ}\\\\4\theta-160^{\circ}=76^{\circ}\\\\4\theta=236^{\circ}\\\\\theta=\boxed{59^{\circ}}[/tex]
