Question

1. which of the following diagrams represent the alternate interior angles theorem and its converse

parallel lines and their angles:
theorems (p→q) and their converse (q→p)
corresponding angles theorem:
if two parallel lines are cut by a transversal, then the corresponding angles are congruent.
corresponding angles converse
if two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.
alternate interior angles theorem:
if two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
alternate interior angles converse
if two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.
alternate exterior angles theorem:
if two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
alternate exterior angles converse
if two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.
consecutive interior angles theorem:
if two parallel lines are cut by a transversal, then the consecutive interior angles are supplementary.
consecutive interior angles converse
if two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel.
transitive property of parallel lines
if two lines are parallel to the same line, then they are parallel to each other.

Explanation:

Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Its converse states that if two lines are cut by a transversal such that alternate interior angles are congruent, then the lines are parallel.
Looking at the options:

• In the first option, ∠3 and ∠5 being supplementary is related to consecutive interior angles (not alternate interior angles).
• In the second option, ∠1 and ∠8 are alternate exterior angles (not alternate interior angles).
• In the third option, ∠4 and ∠5 are consecutive interior angles (not alternate interior angles).
• In the fourth option, ∠2 and ∠6 are alternate interior angles. If \(j\parallel k\) (theorem - parallel lines give congruent alternate interior angles) and if ∠2 and ∠6 are congruent then \(j\parallel k\) (converse - congruent alternate interior angles imply parallel lines).

Answer:

The fourth diagram (with ∠2 and ∠6) represents the Alternate Interior Angles Theorem and its Converse.