1. **State the problem:** We are given a circle with a tangent line touching it at one point. The angle between the tangent line and the radius to the point of tangency is $x^\circ$. Another interior angle near the tangent point is $125^\circ$, and an exterior angle outside the tangent line is $22^\circ$. We need to write an equation to solve for $x$ and then find its value. 2. **Recall the tangent-secant angle theorem:** The angle between a tangent and a chord through the point of contact is equal to half the measure of the intercepted arc. Here, the angle $22^\circ$ is equal to half the difference between $125^\circ$ and $x^\circ$. 3. **Write the equation:** $$22 = \frac{1}{2} (125 - x)$$ 4. **Solve for $x$:** Multiply both sides by 2: $$2 \times 22 = 2 \times \frac{1}{2} (125 - x)$$ $$44 = \cancel{2} \times \frac{1}{\cancel{2}} (125 - x)$$ $$44 = 125 - x$$ 5. **Isolate $x$:** $$44 - 125 = 125 - x - 125$$ $$-81 = -x$$ 6. **Multiply both sides by $-1$ to solve for $x$:** $$-1 \times (-81) = -1 \times (-x)$$ $$81 = x$$ **Final answer:** $$x = 81^\circ$$