The problem involves two parallel lines cut by a transversal, with angle $x^\circ$ and another angle labeled as $108^\circ$. The goal is to determine the value of $x^\circ$ and identify the correct reasoning from the options provided.

Solution:

1. Same-Side Exterior Angles Theorem: When two parallel lines are cut by a transversal, the exterior angles on the same side of the transversal are supplementary. This means that their measures add up to $180^\circ$.

Thus: $x^\circ + 108^\circ = 180^\circ$

Solving for $x^\circ$: $x^\circ = 180^\circ - 108^\circ = 72^\circ$

Correct Statement:

The correct reasoning is that same-side exterior angles are supplementary. Therefore, the answer is:

• C. $m \angle x = 72^\circ$ because same-side exterior angles are supplementary.

Would you like more details on why this rule applies, or do you have any other questions? Here are some related questions:

1. What is the definition of corresponding angles in parallel lines?
2. How can you identify alternate interior angles?
3. What is the relationship between vertical angles?
4. Can the transversal line form congruent angles in some scenarios?
5. How are supplementary angles defined in terms of degrees?

Tip: Always remember that angles on the same side of a transversal line, outside the parallel lines, are supplementary, summing up to 180°.