\( \left. \begin{array} { l } { \hat { \alpha } = 2 x + 8 ^ { \circ } } \\ { \hat { \varepsilon } = 4 x - 62 ^ { \circ } } \end{array} \right. \)

To solve for \( x \) in the given equations, we can use the fact that the angles \( \hat{\alpha} \) and \( \hat{\varepsilon} \) are likely related in some way, possibly as supplementary angles, complementary angles, or angles in a geometric figure. Given: \[ \hat{\alpha} = 2x + 8^\circ \] \[ \hat{\varepsilon} = 4x - 62^\circ \] Let's assume that these angles are supplementary, meaning they add up to \( 180^\circ \): \[ \hat{\alpha} + \hat{\varepsilon} = 180^\circ \] Substituting the expressions for \( \hat{\alpha} \) and \( \hat{\varepsilon} \): \[ (2x + 8^\circ) + (4x - 62^\circ) = 180^\circ \] Now, combine like terms: \[ 2x + 4x + 8^\circ - 62^\circ = 180^\circ \] \[ 6x - 54^\circ = 180^\circ \] Next, add \( 54^\circ \) to both sides: \[ 6x = 180^\circ + 54^\circ \] \[ 6x = 234^\circ \] Now, divide by \( 6 \): \[ x = \frac{234^\circ}{6} = 39^\circ \] Now that we have \( x \), we can find the values of \( \hat{\alpha} \) and \( \hat{\varepsilon} \): \[ \hat{\alpha} = 2(39^\circ) + 8^\circ = 78^\circ + 8^\circ = 86^\circ \] \[ \hat{\varepsilon} = 4(39^\circ) - 62^\circ = 156^\circ - 62^\circ = 94^\circ \] Thus, the angles are: \[ \hat{\alpha} = 86^\circ, \quad \hat{\varepsilon} = 94^\circ \] Finally, we can verify that they are supplementary: \[ \hat{\alpha} + \hat{\varepsilon} = 86^\circ + 94^\circ = 180^\circ \] Therefore, the solution is consistent. The value of \( x \) is: \[ \boxed{39^\circ} \]