In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the opposite angles are supplementary. This means that:
\[ m\angle C + m\angle E = 180^\circ. \]
Given that \( m\angle C = 9x^\circ \) and \( m\angle E = 7x + 4^\circ \), we can set up the equation:
\[ 9x + (7x + 4) = 180. \]
Simplifying this, we have:
\[ 9x + 7x + 4 = 180 \] \[ 16x + 4 = 180 \] \[ 16x = 176 \] \[ x = 11. \]
Now we can find the measure of \( \angle E \):
\[ m\angle E = 7x + 4 = 7(11) + 4 = 77 + 4 = 81^\circ. \]
Thus, the measure of \( \angle E \) is \( \boxed{81^\circ} \).
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