1. **Problem Statement:** We have two isosceles triangles, ABC and DEF. In triangle ABC, sides AB and AC are equal, and side BC is labeled as $(x - 4)$. In triangle DEF, sides DF and DE are equal, and side EF is 19. We need to find the value of $x$. 2. **Key Concept:** In isosceles triangles, the sides opposite equal angles are equal. Since both triangles are isosceles with the same pattern of equal sides, and the problem implies a relationship between side BC and side EF, we can set their lengths equal if the triangles are congruent or similar. 3. **Setting up the equation:** Since $BC = x - 4$ and $EF = 19$, and these sides correspond, we have: $$x - 4 = 19$$ 4. **Solving for $x$:** Add 4 to both sides: $$x = 19 + 4$$ $$x = 23$$ 5. **Answer:** The value of $x$ is 23. This solution assumes the triangles are congruent or similar such that corresponding sides are equal, which is a common approach in geometry problems involving isosceles triangles with marked equal sides and given side lengths.