Since triangles \( \Delta ABC \) and \( \Delta DEF \) are congruent, their corresponding angles are equal. Thus, we can set up the equation:
\[ m \angle A = m \angle D \]
Substituting the given expressions for the angles, we have:
\[ x + 36 = 2x \]
Now, we can solve for \( x \) by isolating it on one side of the equation:
\[ x + 36 = 2x \] \[ 36 = 2x - x \] \[ 36 = x \]
Now that we have \( x \), we can find the measure of \( m \angle A \) and \( m \angle D \):
\[ m \angle A = x + 36 = 36 + 36 = 72 \]
\[ m \angle D = 2x = 2(36) = 72 \]
Therefore, the measures of the angles are:
\[ m \angle A = 72^\circ \quad \text{and} \quad m \angle D = 72^\circ \]
This confirms that both angles are equal, as expected in congruent triangles.