Question

write an indirect proof of the converse of the corresponding angles theorem following the outline below. given ∠1 ≅ ∠2 prove: l∥n assume l is not parallel to n. then it is possible to construct a line m that is parallel to l and passes through the intersection of lines t and n. since line m and line n intersect and are not parallel to each other, m∠3 > 0 (do not include the degree symbol in your answer.) by the alternate interior angles theorem, the angle formed by combining angles 2 and 3 is complementary to ∠4, or equivalently, m∠2 + m∠3 = m∠4

Explanation:

Step1: Recall Alternate Interior Angles Theorem

When a transversal crosses parallel lines, alternate interior angles are equal. Here, \( m \parallel l \), so by Alternate Interior Angles Theorem, \( \angle 1=\angle 4 \) (since \( \angle 1 \) and \( \angle 4 \) are alternate interior angles for lines \( m \) and \( l \) with transversal \( t \)). But we know \( \angle 1 = \angle 2 \), so \( \angle 2=\angle 4 \).

Step2: Analyze the angle sum

We have the equation \( m\angle 2 + m\angle 3=m\angle 4 \) (from the given). Substitute \( \angle 2=\angle 4 \) into it: \( m\angle 2 + m\angle 3=m\angle 2 \). Subtract \( m\angle 2 \) from both sides: \( m\angle 3 = 0 \). But we assumed \( m\angle 3>0 \), which is a contradiction. So our initial assumption ( \( l \) not parallel to \( n \)) is false, hence \( l\parallel n \).

Answer:

The indirect proof shows that assuming \( l \) is not parallel to \( n \) leads to a contradiction (\( m\angle 3 = 0 \) vs \( m\angle 3>0 \)), so \( l\parallel n \). The key steps use the Alternate Interior Angles Theorem and angle - sum relations to derive the contradiction.