Since quadrilateral CDEF is inscribed in circle A, we can use the fact that the opposite angles of an inscribed quadrilateral are supplementary. Therefore, the sum of the measures of angles C and E must be \(180^\circ\):
\[ m\angle C + m\angle E = 180^\circ \]
Given:
- \(m\angle C = 9x^\circ\)
- \(m\angle E = 7x + 4^\circ\)
We substitute these into the equation:
\[ 9x + (7x + 4) = 180 \]
Combining like terms gives:
\[ 16x + 4 = 180 \]
Next, we isolate \(16x\) by subtracting 4 from both sides:
\[ 16x = 180 - 4 \] \[ 16x = 176 \]
Now, divide both sides by 16 to find \(x\):
\[ x = \frac{176}{16} = 11 \]
Now that we have the value of \(x\), we can find \(m\angle E\):
\[ m\angle E = 7x + 4 \] \[ m\angle E = 7(11) + 4 = 77 + 4 = 81^\circ \]
Thus, the measure of \( \angle E \) is:
\[ \boxed{81^\circ} \]
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