The Alternate Interior Angles Theorem states that, when two parallel lines are cut by a transversal , the resulting alternate interior angles are congruent . So, in the figure below, if k ∥ l , then ∠ 2 ≅ ∠ 8 and ∠ 3 ≅ ∠ 5 .
Proof. Since k ∥ l , by the Corresponding Angles Postulate , ∠ 1 ≅ ∠ 5 . Therefore, by the definition of congruent angles , m ∠ 1 = m ∠ 5 . Since ∠ 1 and ∠ 2 form a linear pair , they are supplementary , so m ∠ 1 + m ∠ 2 = 180 ° . Also, ∠ 5 and ∠ 8 are supplementary, so m ∠ 5 + m ∠ 8 = 180 ° . Substituting m ∠ 1 for m ∠ 5 , we get m ∠ 1 + m ∠ 8 = 180 ° . Subtracting m ∠ 1 from both sides, we have m ∠ 8 = 180 ° − m ∠ 1 = m ∠ 2 . Therefore, ∠ 2 ≅ ∠ 8 . You can prove that ∠ 3 ≅ ∠ 5 using the same method. The converse of this theorem is also true; that is, if two lines k and l are cut by a transversal so that the alternate interior angles are congruent, then k ∥ l . |