In this problem, two parallel lines are cut by a transversal, and we are given the measure of two angles: angle 1 and an angle adjacent to its alternate exterior angle.

Key concepts:

• Alternate exterior angles are congruent when two parallel lines are cut by a transversal.
• Adjacent angles on a straight line are supplementary, meaning they sum to $180^\circ$.

Step-by-step solution:

1. Identify the angles:

   • Let angle 1 be $(4x + 28)^\circ$.
   • The angle adjacent to the alternate exterior angle of angle 1 is $(14x + 8)^\circ$.

2. Since these two angles are adjacent on a straight line, they are supplementary, meaning their sum is $180^\circ$.

3. Set up the equation: $(4x + 28)^\circ + (14x + 8)^\circ = 180^\circ$

4. Combine like terms: $(4x + 14x) + (28 + 8) = 180$ $18x + 36 = 180$

5. Solve for $x$: $18x = 180 - 36$ $18x = 144$ $x = \frac{144}{18} = 8$

Conclusion:

The value of $x$ is $8$.

Would you like further explanation, or do you have any other questions?

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Here are 5 related questions:

1. What are alternate interior angles and their properties?
2. How do we find the value of other angles in a transversal problem?
3. Can angles on the same side of a transversal be equal?
4. What are consecutive interior angles and their relationships?
5. How do you solve equations with angles in terms of $x$?

Tip: In geometry, supplementary angles always add up to $180^\circ$, regardless of their specific types.