Three lines $m$, $n$, and $\ell$ lie in a plane such that no two are parallel. Lines $m$ and $n$ meet at an acute angle of $14^{\circ}$, and lines $m$ and $\ell$ meet at an acute angle of $20^{\circ}$. Find, in degrees, the sum of all possible acute angles formed by lines $n$ and $\ell$.

[i]Ray Li[/i]
1. Given three lines \( m \), \( n \), and \( \ell \) in a plane such that no two are parallel. Lines \( m \) and \( n \) meet at an acute angle of \( 14^\circ \), and lines \( m \) and \( \ell \) meet at an acute angle of \( 20^\circ \).

2. We need to find the sum of all possible acute angles formed by lines \( n \) and \( \ell \).

3. Consider the angles formed by the intersection of lines \( n \) and \( \ell \). Since \( m \) intersects both \( n \) and \( \ell \), we can use the given angles to determine the possible angles between \( n \) and \( \ell \).

4. Let \(\theta\) be the acute angle between lines \( n \) and \( \ell \). We have two cases to consider:
   - Case 1: \( \ell \) is positioned such that the angle between \( n \) and \( \ell \) is the sum of the angles between \( m \) and \( n \) and between \( m \) and \( \ell \).
   - Case 2: \( \ell \) is positioned such that the angle between \( n \) and \( \ell \) is the difference of the angles between \( m \) and \( n \) and between \( m \) and \( \ell \).

5. For Case 1, the angle between \( n \) and \( \ell \) is:
   \[
   \theta_1 = 14^\circ + 20^\circ = 34^\circ
   \]

6. For Case 2, the angle between \( n \) and \( \ell \) is:
   \[
   \theta_2 = 20^\circ - 14^\circ = 6^\circ
   \]

7. Both \( \theta_1 \) and \( \theta_2 \) are acute angles, so we sum these angles to find the total sum of all possible acute angles formed by lines \( n \) and \( \ell \):
   \[
   \theta_1 + \theta_2 = 34^\circ + 6^\circ = 40^\circ
   \]

Conclusion:
The sum of all possible acute angles formed by lines \( n \) and \( \ell \) is \( \boxed{40} \).