1. **State the problem:** We need to find the value of angle $x$ in a triangle where two sides and one angle are given. 2. **Given:** Side lengths 5.7, 7, and 7; angles $x^\circ$ (bottom-left vertex) and $66^\circ$ (right vertex). 3. **Use the Triangle Angle Sum Rule:** The sum of interior angles in any triangle is $180^\circ$. 4. **Calculate the third angle:** Let the third angle be $y$. Then, $$x + 66 + y = 180$$ 5. **Use the Law of Cosines to find $y$:** Since sides opposite to angles $x$, $66^\circ$, and $y$ are 5.7, 7, and 7 respectively, and the side opposite $66^\circ$ is 5.7, we can find $y$ using Law of Cosines: $$\cos(y) = \frac{7^2 + 7^2 - 5.7^2}{2 \times 7 \times 7}$$ 6. **Calculate:** $$\cos(y) = \frac{49 + 49 - 32.49}{98} = \frac{65.51}{98} \approx 0.6683$$ 7. **Find $y$:** $$y = \cos^{-1}(0.6683) \approx 48.1^\circ$$ 8. **Find $x$:** $$x = 180 - 66 - 48.1 = 65.9^\circ$$ **Final answer:** $$x \approx 65.9^\circ$$