Question

what is the converse of the alternate interior angles theorem? if p and q are parallel, and r is a transversal, then angle d is congruent to angle e.

Explanation:

Step1: Recall the converse of a conditional statement

A conditional statement is in the form "If \( A \), then \( B \)". The converse of this statement is "If \( B \), then \( A \)".

Step2: Identify \( A \) and \( B \) from the given theorem

In the given alternate interior angles theorem: "If \( p \) and \( q \) are parallel, and \( r \) is a transversal, then angle \( D \) is congruent to angle \( E \)". Here, \( A \) is " \( p \) and \( q \) are parallel, and \( r \) is a transversal" and \( B \) is "angle \( D \) is congruent to angle \( E \)".

Step3: Form the converse

Using the converse structure (If \( B \), then \( A \)), we get: If angle \( D \) is congruent to angle \( E \), and \( r \) is a transversal, then \( p \) and \( q \) are parallel.

Answer:

If angle \( D \) is congruent to angle \( E \), and \( r \) is a transversal, then \( p \) and \( q \) are parallel.