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To solve for \(\angle E\) in the given geometric figure, we need to analyze the angles provided and use the properties of angles and triangles.

1. In the figure, \(\angle DCE = 42^\circ\).
2. Let's denote the angles at point C as follows:
    - \(\angle BCA = \varphi + \rho\)
    - \(\angle ACD = \varphi - 3^\circ\)

Since line \(BC\) and line \(DE\) intersect at point C, and \(\angle BCA\) and \(\angle ACD\) form a straight line, the sum of these angles should be \(180^\circ\).

Using the straight-line property:
\[
\angle BCA + \angle ACD = 180^\circ
\]

Substitute the given angles:
\[
(\varphi + \rho) + (\varphi - 3^\circ) = 180^\circ
\]

Simplify:
\[
2\varphi + \rho - 3^\circ = 180^\circ
\]

Now, note that \(\angle DCE = 42^\circ\). Since \(\angle BCA\) and \(\angle ACD\) form a linear pair with \(\angle DCE\), we can determine \(\angle EDC\) using the fact that the sum of the interior angles of a triangle is \(180^\circ\). Therefore, in triangle \(CDE\):

\[
\angle EDC + \angle DCE + \angle DEC = 180^\circ
\]

We know \(\angle DCE = 42^\circ\), and the other angle \(\angle DEC\) is the angle we need to find. Hence, let’s denote \(\angle DEC\) as \(x\).

\[
\angle EDC + 42^\circ + x = 180^\circ
\]

Since \(\angle EDC\) is formed by the exterior angles to \(\angle DCE\) and \(\angle DEC\), let's use the property that the sum of the exterior angle of a triangle is equal to the sum of the opposite interior angles. 

Given:
\[
\angle EDC = \angle BCA = \varphi + \rho
\]

Thus,
\[
(\varphi + \rho) + 42^\circ + x = 180^\circ
\]

\[
x = 180^\circ - 42^\circ - (\varphi + \rho)
\]

Now substitute the values of \(\varphi + \rho\) and \(\varphi - 3^\circ\) we found earlier.

Since:
\[
2\varphi + \rho - 3^\circ = 180^\circ \Rightarrow \rho = 183^\circ - 2\varphi
\]

Substitute \(\rho\):
\[
x = 180^\circ - 42^\circ - (183^\circ - 2\varphi) + \varphi
\]
\[
x = 180^\circ - 42^\circ - 183^\circ + 2\varphi - \varphi
\]
\[
x = -45^\circ + \varphi
\]

Thus, since there must be a correct interpretation of the angles which sums up and make sense geometrically and algebraically, which is achievable only when all angles provided consistently confirm the figure.

So, the value of \(\angle E\):
\[
x = 180^\circ - 42^\circ - (BCA)
= 180^\circ - 42^\circ - (\varphi + \rho) = 45^\circ \text{(assumed consistent approach)}\)

In summary:
**\(\angle E = 180^\circ - \angle DCE = 180^\circ - 42^\circ = 138^\circ (in generic terms if visual figure confirmation).**

Would you like further details or have any questions? 

Here are 8 relative questions:

1. What are the properties of exterior angles in a triangle?
2. How can you use the sum of angles in a triangle to solve for unknown angles?
3. What is the relationship between supplementary and linear pair of angles?
4. How do you interpret angle bisectors in geometric figures?
5. What is the significance of vertical angles in intersecting lines?
6. How can angle sum property of triangles help in geometric proofs?
7. What are the strategies to solve problems involving exterior and interior angles?
8. How do you solve for unknown variables when angles are given in algebraic form?

**Tip:** Always double-check each step and verify angle relationships when solving geometric problems to ensure accuracy.

Explore how to solve for angle E in geometric figures using triangle properties and angle relationships. This problem involves analyzing angles in triangles, utilizing the angle sum property, and applying linear pair concepts. Suitable for high school students.

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