To find the value of [tex]\( x \)[/tex] given that the angle measures are [tex]\( (4x + 28)^\circ \)[/tex] and the adjacent angle to the alternate exterior angle to this first angle is [tex]\( (14x + 8)^\circ \)[/tex], we need to consider the properties of angles formed by parallel lines and a transversal.
When parallel lines are cut by a transversal, alternate exterior angles are equal. Knowing this, we can set up our equation.
However, in this problem, these angles are described with the angles adjacent to the alternate exterior angle forming a linear pair with angle 1.
1. Identify the angles: The two given angles are [tex]\((4x + 28)^\circ\)[/tex] and [tex]\((14x + 8)^\circ\)[/tex].
2. Form the equation: Since these angles are adjacent (forming a linear pair), the sum of these angles must be [tex]\(180^\circ\)[/tex].
Thus, we write the equation:
[tex]\[ (4x + 28) + (14x + 8) = 180 \][/tex]
3. Combine like terms:
[tex]\[ 4x + 14x + 28 + 8 = 180 \][/tex]
[tex]\[ 18x + 36 = 180 \][/tex]
4. Isolate [tex]\( x \)[/tex]:
[tex]\[ 18x + 36 = 180 \][/tex]
[tex]\[ 18x = 180 - 36 \][/tex]
[tex]\[ 18x = 144 \][/tex]
[tex]\[ x = \frac{144}{18} \][/tex]
[tex]\[ x = 8 \][/tex]
So, the value of [tex]\( x \)[/tex] is 8. Therefore, the correct answer is:
[tex]\[ \boxed{8} \][/tex]
