1. **State the problem:** Given two congruent triangles $\triangle ABC \cong \triangle FDE$, find the values of $x$ and $y$ given the angles: - $\angle C = 108^\circ$ - $\angle B = 48^\circ$ - $\angle F = (2x - y)^\circ$ - $\angle D = y^\circ$ 2. **Recall the property of congruent triangles:** Corresponding angles of congruent triangles are equal. So, $$\angle A = \angle F, \quad \angle B = \angle D, \quad \angle C = \angle E.$$ 3. **Use the given angles:** - Since $\angle B = 48^\circ$ and $\angle D = y^\circ$, we have $$y = 48.$$ 4. **Find $\angle A$:** The sum of angles in a triangle is $180^\circ$, so for $\triangle ABC$: $$\angle A + \angle B + \angle C = 180^\circ$$ $$\angle A + 48 + 108 = 180$$ $$\angle A + 156 = 180$$ $$\angle A = 180 - 156 = 24.$$ 5. **Use congruence for $\angle F$:** Since $\angle A = \angle F$, $$\angle F = 24^\circ.$$ 6. **Set up equation for $x$ and $y$:** Given $\angle F = (2x - y)^\circ$, substitute $y=48$ and $\angle F=24$: $$2x - 48 = 24.$$ 7. **Solve for $x$:** $$2x = 24 + 48 = 72$$ $$x = \frac{\cancel{2}x}{\cancel{2}} = \frac{72}{2} = 36.$$ **Final answers:** $$x = 36, \quad y = 48.$$