1. **Stating the problem:** We have a circle tangent to two rays forming a 70° angle at point Q. The upper tangent touches the circle at R, and the lower tangent touches the circle at P. The arc or side on the right is labeled with the expression $83x + 1$. We need to find the value of $x$. 2. **Understanding the geometry:** The two rays form an angle of 70° at Q. The circle is tangent to both rays, so the tangents from Q to the circle at points P and R create equal angles with the radius at the points of tangency. 3. **Key property:** The angle between the two tangents from a point outside a circle equals the difference between 180° and the measure of the intercepted arc between the points of tangency. 4. **Formula:** If the angle between the tangents at Q is $\theta$, and the intercepted arc between P and R is $m$, then: $$\theta = 180^\circ - m$$ 5. **Apply the formula:** Given $\theta = 70^\circ$, then: $$70 = 180 - (83x + 1)$$ 6. **Solve for $x$:** $$83x + 1 = 180 - 70$$ $$83x + 1 = 110$$ 7. **Isolate $x$:** $$83x = 110 - 1$$ $$83x = 109$$ 8. **Divide both sides by 83:** $$x = \frac{109}{83}$$ 9. **Simplify fraction if possible:** 109 and 83 have no common factors other than 1, so: $$x = \frac{109}{83} \approx 1.313$$ **Final answer:** $$x = \frac{109}{83}$$