1. **Problem statement:** We need to find the value of angle $x$ in a right triangle with a smaller isosceles triangle nested inside it. The smaller triangle has an angle of $48^\circ$, and the larger triangle has a right angle ($90^\circ$) and an angle $x^\circ$ adjacent to it. 2. **Key facts and formulas:** - The sum of angles in any triangle is $180^\circ$. - In an isosceles triangle, two sides are equal, so the angles opposite those sides are equal. 3. **Analyze the smaller isosceles triangle:** - One angle is $48^\circ$. - Since it is isosceles, the other two angles are equal. - Let each of these equal angles be $y$. 4. **Sum of angles in the smaller triangle:** $$48 + y + y = 180$$ $$48 + 2y = 180$$ $$2y = 180 - 48$$ $$2y = 132$$ $$y = \frac{132}{2} = 66$$ 5. **Interpretation:** - The two equal angles in the smaller triangle are each $66^\circ$. - The angle $x$ in the larger triangle corresponds to one of these $66^\circ$ angles because it is adjacent to the right angle and shares the side with the smaller triangle. 6. **Final answer:** $$x = 66^\circ$$ Thus, the value of $x$ is $66$ degrees.