1. **State the problem:** We have two triangles sharing a common base with congruent sides on the left. The upper triangle has an angle of 68° and side length 77 on the right, while the lower triangle has an angle of 32° and side length $11x - 33$ on the right. We need to find the range of values for $x$ such that the triangles' side lengths and angles are consistent. 2. **Use the Law of Sines:** For triangles with two sides and an included angle, the Law of Sines relates sides and opposite angles: $$\frac{a}{\sin A} = \frac{b}{\sin B}$$ Here, the sides opposite the given angles are 77 and $11x - 33$ respectively. 3. **Set up the ratio:** $$\frac{77}{\sin 68^\circ} = \frac{11x - 33}{\sin 32^\circ}$$ 4. **Solve for $x$:** Multiply both sides by $\sin 32^\circ$: $$77 \cdot \frac{\sin 32^\circ}{\sin 68^\circ} = 11x - 33$$ Add 33 to both sides: $$77 \cdot \frac{\sin 32^\circ}{\sin 68^\circ} + 33 = 11x$$ Divide both sides by 11: $$x = \frac{77 \cdot \frac{\sin 32^\circ}{\sin 68^\circ} + 33}{11}$$ 5. **Calculate the sine values:** $\sin 32^\circ \approx 0.5299$ $\sin 68^\circ \approx 0.9272$ 6. **Evaluate the expression:** $$x = \frac{77 \cdot \frac{0.5299}{0.9272} + 33}{11} = \frac{77 \cdot 0.5715 + 33}{11} = \frac{44.0 + 33}{11} = \frac{77}{11} = 7$$ 7. **Check the side length $11x - 33$ for positivity:** Since side lengths must be positive: $$11x - 33 > 0$$ $$11x > 33$$ $$x > 3$$ 8. **Determine the range for $x$:** From the Law of Sines, $x = 7$ exactly for the triangles to be similar with given angles and sides. From the positivity condition, $x > 3$. Therefore, the valid range for $x$ is: $$3 < x < 7$$ **Final answer:** $$3 < x < 7$$