Converse Corresponding Angles Theorem

This theorem can be proven by an indirect proof. Let l_1 and l_2 be two lines intersected by a transversal line l_3 forming corresponding congruent angles ∠1 and ∠2.

Since the goal is to prove that l_1 is parallel to l_2, it will be temporarily assumed that l_1 and l_2 are not parallel. Temporary Assumption l_1 ∦ l_2 By the Parallel Postulate, there exists a line n parallel to l_2 that passes through the point of intersection between l_1 and l_3. This line forms ∠3 and ∠4.

By the Angle Addition Postulate, m∠1 is equal to the sum of m∠3 and m∠4. m∠1=m∠3+m∠4 Since n and l_2 are parallel lines that are cut by a transversal, by the Corresponding Angles Theorem, ∠3 and ∠2 are congruent. By the definition of congruence, these angles have the same measure. ∠3 ≅ ∠2 ⇕ m∠3 = m∠2 By the Substitution Property of Equality, m∠2 can be substituted for m∠3 into the equation for m∠1. m∠1=m∠3+m∠4 ↓ m∠1= m∠2+m∠4 From the above equation and since m∠ 4 is a positive number, it can be concluded that m∠1 is greater than m∠2. m∠1>m∠2 This contradicts the given fact that ∠1 and ∠2 are congruent. The contradiction came from assuming that l_1 and l_2 are not parallel lines. Therefore, l_1 and l_2 must be parallel lines.