1. **State the problem:** We have an isosceles right triangle with one angle measuring 55.5° and a side opposite this angle labeled as $2x + 7$. We need to find the value of $x$. 2. **Recall properties of isosceles right triangles:** In an isosceles right triangle, the two legs are equal, and the angles are 45°, 45°, and 90°. However, here one angle is 55.5°, so the triangle is not a standard isosceles right triangle. The problem states it is isosceles, so two sides are equal, and the angles opposite those sides are equal. 3. **Use the triangle angle sum rule:** The sum of angles in any triangle is 180°. Let the two equal angles be $55.5^\circ$ and $\theta$. Since the triangle is isosceles, the two equal sides are opposite the two equal angles. Given one angle is 55.5°, the other equal angle must also be 55.5°. Calculate the third angle: $$\text{Third angle} = 180^\circ - 55.5^\circ - 55.5^\circ = 69^\circ$$ 4. **Set up the sides:** The side opposite the 55.5° angle is $2x + 7$. Since the triangle is isosceles, the side opposite the other 55.5° angle is also $2x + 7$. 5. **Use the Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ Let the side opposite the 69° angle be $c$. 6. **Express the side opposite 69° in terms of $2x + 7$:** $$\frac{2x + 7}{\sin 55.5^\circ} = \frac{c}{\sin 69^\circ}$$ 7. **Since the triangle is isosceles, the two sides opposite 55.5° are equal, so no new equation for $x$ here. We need more information or a relation involving $x$.** 8. **Assuming the side opposite 69° is the hypotenuse, and the triangle is right-angled at the 69° angle, which contradicts the problem statement.** 9. **Alternatively, if the triangle is isosceles with two equal sides $2x + 7$, and the angle between them is 69°, then use the Law of Cosines to find the third side. But since the problem only asks for $x$, and no other side length is given, we cannot solve for $x$ without additional information.** **Conclusion:** The problem as stated lacks sufficient information to solve for $x$. Please provide the length of another side or clarify the triangle's properties.