The answer is: ∠I = ∠L (Vertical angles), ∠HJI = ∠LKJ (Alternate interior angles), ∠I = ∠LKJ (SAS-Postulate), ΔHIJ ≅ ΔALK (AA Similarity).
The given information is as follows:
Triangle HIJKL is given, with HI parallel to KL.
We need to prove that triangle HIJ is congruent to triangle ALK.
Steps to solve:
1. Identify vertical angles: Since HI is parallel to KL, we know that angles I and L are vertical angles. Vertical angles are congruent by definition, so we can write ∠I = ∠L.
2. Identify alternate interior angles: Angles HJI and LKJ are alternate interior angles formed by transversal JK intersecting lines HI and KL. Alternate interior angles are congruent by definition, so we can write ∠HJI = ∠LKJ.
3. Apply the transitive property of congruence: Since we have ∠I = ∠L and ∠HJI = ∠LKJ, we can use the transitive property of congruence to conclude that ∠I = ∠LKJ.
4. Apply AA Similarity: Now that we have two pairs of congruent angles (∠I = ∠L and ∠HJI = ∠LKJ), we can apply the AA Similarity postulate to triangles HIJ and ALK. This postulate states that if two triangles have two pairs of congruent angles, then the triangles are similar.
Therefore, triangles HIJ and ALK are congruent by AA Similarity.
The answer is:
1. ∠I = ∠L (Vertical angles)
2. ∠HJI = ∠LKJ (Alternate interior angles)
3. ∠I = ∠LKJ (SAS-Postulate)
4. ΔHIJ ≅ ΔALK (AA Similarity)