To solve the problem, we need to use the properties of angles formed by a transversal intersecting two parallel lines. The problem states that angle 1 measures \((4x + 28)^\circ\) and the angle adjacent to the alternate exterior angle with angle 1 measures \((14x + 8)^\circ\).
First, let's recall that alternate exterior angles are congruent when two parallel lines are intersected by a transversal. In other words, the alternate exterior angle to angle 1 should have the same measure as angle 1. However, we are given another angle adjacent to this alternate exterior angle, which means it will add up to these angles summing to 180 degrees along a straight line.
Let's set up our equations step-by-step:
1. Define angle 1 as \((4x + 28)^\circ\).
2. Denote the angle adjacent to the alternate exterior angle as \((14x + 8)^\circ\).
Since these angles add up to form a straight line (180 degrees):
[tex]\[
(4x + 28) + (14x + 8) = 180
\][/tex]
Let's solve this step-by-step:
Combine like terms:
[tex]\[
4x + 14x + 28 + 8 = 180
\][/tex]
[tex]\[
18x + 36 = 180
\][/tex]
Next, isolate the variable by subtracting 36 from both sides:
[tex]\[
18x + 36 - 36 = 180 - 36
\][/tex]
[tex]\[
18x = 144
\][/tex]
Finally, divide both sides by 18 to solve for \(x\):
[tex]\[
x = \frac{144}{18}
\][/tex]
[tex]\[
x = 8
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] is [tex]\(2\)[/tex].
