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Answer :
Final answer:
The theorem that cannot be used to prove two lines are parallel is D If two lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the lines are parallel.
Explanation:
In this question, we need to determine which theorem cannot be used to prove two lines are parallel. Let's go through each theorem:
A: If two lines are cut by a transversal so that a pair of same side interior angles are supplementary, then the lines are parallel. This is a valid theorem that can be used to prove parallel lines.
B: If two lines are cut by a transversal so that a pair of corresponding angles are congruent, then the lines are parallel. This is also a valid theorem that can be used to prove parallel lines.
C: If two lines are cut by a transversal so that a pair of vertical angles are congruent, then the lines are parallel. This is a valid theorem that can be used to prove parallel lines.
D: If two lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the lines are parallel. This theorem cannot be used to prove parallel lines, as alternate exterior angles being congruent does not guarantee parallel lines.
Therefore, the theorem that cannot be used to prove parallel lines is D.
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