The diagram shows a pair of parallel lines intersected by a transversal, with one given angle of $108^\circ$ and the other angle labeled $x^\circ$. Below the diagram, there are multiple answer choices asking to identify the correct justification for the angle $x^\circ = 72^\circ$.

Solution:

Since the two lines are parallel, several angle relationships apply. The angle next to the $108^\circ$ is a linear pair, meaning they add up to $180^\circ$. So,$x + 108^\circ = 180^\circ$ $x = 180^\circ - 108^\circ = 72^\circ$

Therefore, $x = 72^\circ$.

Correct Answer Explanation:

The correct reason for $m\angle x = 72^\circ$ is based on vertical angles. Vertical angles are congruent when two lines intersect.

Hence, the correct option is D: $m\angle x = 72^\circ$ because vertical angles are congruent.

Would you like more details or have any questions about this explanation?

Here are 5 related questions:

1. What are alternate interior angles, and how do they apply to parallel lines?
2. How do corresponding angles behave when two lines are cut by a transversal?
3. Can you explain how same-side exterior angles are related in parallel lines?
4. What are linear pairs of angles, and why are they supplementary?
5. How can you determine if two angles are congruent using geometric properties?

Tip: Always look for relationships between angles such as vertical, corresponding, and alternate interior angles when dealing with parallel lines.