To solve the problem, we first need to identify the relationships between the angles formed by the two parallel lines and the transversal.
Given:
- Angle 1 measures \( (4x + 28)^\circ \)
- The angle adjacent to the alternate exterior angle with angle 1 measures \( (14x + 8)^\circ \).
Since angle 1 and the adjacent angle are corresponding angles created by the transversal crossing the parallel lines, they are equal. Therefore, we can set up the equation:
\[ 4x + 28 = 14x + 8 \]
Now, we will isolate \( x \):
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Subtract \( 4x \) from both sides: \[ 28 = 10x + 8 \]
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Subtract 8 from both sides: \[ 20 = 10x \]
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Divide both sides by 10: \[ x = 2 \]
Thus, the value of \( x \) is \( \boxed{2} \).