When two parallel lines are cut by a transversal, are the alternate exterior angles always congruent?

Introduction / Context:
In Euclidean geometry, when two parallel lines are intersected by a third line, called a transversal, several special angle pairs are formed. These include corresponding angles, alternate interior angles and alternate exterior angles. This question tests your understanding of the property of alternate exterior angles when the two lines are parallel.

Given Data / Assumptions:
- There are two distinct straight lines in a plane.
- These two lines are stated to be parallel to each other.
- A transversal line cuts both parallel lines, forming several angles at the points of intersection.
- We focus on alternate exterior angles formed outside the region between the two parallel lines, on opposite sides of the transversal.

Concept / Approach:
In Euclidean geometry, when a transversal intersects two parallel lines, alternate exterior angles are always equal in measure, or congruent. This is closely related to the fact that corresponding angles are equal and that the sum of the angles on a straight line is 180 degrees. Parallelism ensures consistent angle relationships everywhere the transversal crosses.

Step-by-Step Solution:
Step 1: Draw two parallel lines and mark them as line l and line m. Step 2: Draw a transversal that intersects both lines at distinct points. Step 3: Label one exterior angle on the first line above or below the parallel lines. Step 4: Identify the angle on the other line that lies outside the two lines and on the opposite side of the transversal. This is the alternate exterior angle. Step 5: Use the parallel line postulate and properties of corresponding angles to see that these two alternate exterior angles have equal measure. Step 6: Conclude that for parallel lines, alternate exterior angles are congruent in every such configuration.

Verification / Alternative Check:
You can verify this with a protractor on an accurate drawing or using dynamic geometry software. Fix two lines as parallel and draw a transversal at different slopes. Measure one alternate exterior angle and then its partner. No matter how the transversal is placed, as long as the lines remain parallel, the pair of alternate exterior angles will always have the same measure.

Why Other Options Are Wrong:
Option B is wrong because alternate exterior angles are not always supplementary; they are equal in measure when lines are parallel. Supplementary angles are a different angle pair type.
Option C is wrong because it ignores the fundamental theorem about parallel lines and transversals that guarantees equality of these angles.
Option D is wrong because if the lines intersect, they are not parallel and the standard parallel line angle relationships do not hold in the same way.

Common Pitfalls:
Students often confuse alternate exterior angles with alternate interior angles or corresponding angles. Another frequent mistake is assuming that all angle pairs formed by a transversal are supplementary, which is not correct. Recognizing the position (inside or outside) and side of the transversal is key to identifying angle types correctly.

Final Answer:
For two parallel lines cut by a transversal, the alternate exterior angles are always congruent.