Alternate Interior Angles Theoremfigure 8Match the justifications for each step in order to prove the Alternate Interior Angles Theorem. Note that lines l and n are parallel.(2 points)
l∥n
∠4≅∠2
∠6≅∠4
∠2≅∠6
Answer choices:
Given; Transitive property; Corresponding angles; Vertical angles
∠4≅∠2 - Corresponding angles
∠6≅∠4 - Corresponding angles
∠2≅∠6 - Transitive property (since ∠4≅∠2 and ∠6≅∠4)
To prove the Alternate Interior Angles Theorem, we need to match the justifications for each step:
Step 1: Given - line l is parallel to line n. (This is given in the problem statement)
Step 2: Corresponding angles - ∠4 and ∠2 are corresponding angles. (This applies because they are on different parallel lines and on the same side of the transversal line)
Step 3: Corresponding angles - ∠6 and ∠4 are corresponding angles. (This applies because they are on different parallel lines and on the same side of the transversal line)
Step 4: Transitive property - Since ∠4 ≅ ∠2 (corresponding angles) and ∠6 ≅ ∠4 (corresponding angles), then ∠2 ≅ ∠6. (This applies because if two angles are equal to the same angle, they are equal to each other as well)
Therefore, the match for each step is as follows:
Step 1: Given
Step 2: Corresponding angles
Step 3: Corresponding angles
Step 4: Transitive property
To prove the Alternate Interior Angles Theorem, we need to match each step with the correct justification:
Step 1: Given
This is the starting point of the proof, indicating that lines l and n are parallel. This assumption is given to us as a premise.
Step 2: Corresponding angles
The justifications for step 2 is "Corresponding angles". According to the Corresponding Angles Postulate, when two parallel lines are intersected by a transversal, the pairs of corresponding angles are congruent. Here, we are identifying that ∠4 and ∠2 are corresponding angles.
Step 3: Transitive property
The justifications for step 3 is "Transitive property". The transitive property states that if two things are equal to the same thing, then they are equal to each other. In this case, since ∠4 is congruent to ∠2 (given in step 2), and ∠6 is congruent to ∠4 (given), we can conclude that ∠6 is congruent to ∠2.
Step 4: Corresponding angles
The justifications for step 4 is "Corresponding angles". Again, using the Corresponding Angles Postulate, we are identifying that ∠2 and ∠6 are corresponding angles.
Therefore, the correct match for each step in order to prove the Alternate Interior Angles Theorem is:
Step 1: Given
Step 2: Corresponding angles
Step 3: Transitive property
Step 4: Corresponding angles