Question

in the given figure, which lines or segments are parallel? justify your answer. choose the correct answer below. a. (overline{ab}paralleloverline{cd}) by the converse of the alternate interior angles theorem b. (overline{cd}paralleloverline{ef}) by the converse of the alternate exterior angles theorem c. (overline{dc}paralleloverline{ab}) by the converse of the corresponding angles theorem d. (overline{cd}paralleloverline{ef}) by the converse of the same - side interior angles postulate

Explanation:

1. Recall the properties of parallel - line theorems:

• The Converse of the Alternate Interior Angles Theorem states that if alternate interior angles are congruent, then the lines are parallel.
• The Converse of the Alternate Exterior Angles Theorem states that if alternate exterior angles are congruent, then the lines are parallel.
• The Converse of the Corresponding Angles Theorem states that if corresponding angles are congruent, then the lines are parallel.
• The Converse of the Same - Side Interior Angles Postulate states that if same - side interior angles are supplementary, then the lines are parallel.

1. Analyze each option:

• Option A: If two lines are parallel by the Converse of the Alternate Interior Angles Theorem, we need to check for alternate interior angles formed by a transversal.
• Option B: If two lines are parallel by the Converse of the Alternate Exterior Angles Theorem, we need to check for alternate exterior angles formed by a transversal.
• Option C: If two lines are parallel by the Converse of the Corresponding Angles Theorem, we need to check for corresponding angles formed by a transversal. This is a valid way to prove two lines are parallel.
• Option D: The Same - Side Interior Angles Postulate requires angles to be supplementary for the lines to be parallel, not just some general relationship as stated in the option.

Answer:

C. \(\overline{DC}\parallel\overline{AB}\) by the Converse of the Corresponding Angles Theorem