To find the value of \( x \), we first need to establish a relationship between the angles given. Suppose \( \angle 3 \) and \( \angle 4 \) are supplementary angles (meaning they add up to \( 180^\circ \)). We can set up the following equation:
\[ m\angle 3 + m\angle 4 = 180^\circ \]
Substituting the given values:
\[ 124^\circ + (8x + 40) = 180^\circ \]
Now we can simplify and solve for \( x \):
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Combine like terms: \[ 124 + 8x + 40 = 180 \] \[ 164 + 8x = 180 \]
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Isolate the term with \( x \): \[ 8x = 180 - 164 \] \[ 8x = 16 \]
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Divide both sides by 8: \[ x = \frac{16}{8} = 2 \]
Thus, the value of \( x \) is \( \boxed{2} \).