Question

1. state the theorem or postulate that proves ( a parallel b ).

a. converse of the corresponding angles theorem
b. converse of the alternate interior angles theorem
c. converse of the same - side interior angles postulate
d. converse of the alternate exterior angles theorem

Explanation:

• Converse of the Alternate Interior Angles Theorem: This theorem states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the two lines are parallel.
• In the given diagram, the \(y^{\circ}\) angle and the \(65^{\circ}\) angle are alternate interior angles. If \(y = 65\) (assuming they are congruent as per the parallel - line proof context), then by the converse of the alternate interior angles theorem, \(a\parallel b\).
• Converse of the Corresponding Angles Theorem: Requires angles in corresponding positions (not the case here as the given angles are alternate interior).
• Converse of the Same - Side Interior Angles Postulate: Requires the sum of same - side interior angles to be \(180^{\circ}\) (not relevant as the given angles are not same - side interior).
• Converse of the Alternate Exterior Angles Theorem: Deals with angles outside the two lines (not the case here as the given angles are between the two lines \(a\) and \(b\)).

Answer:

B. Converse of the Alternate Interior Angles Theorem