Alternate Interior Angles Theorem
figure 8
Match the justifications for each step in order to prove the Alternate Interior Angles Theorem. Note that lines l and n are parallel.
(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
∠6≅∠4
∠2≅∠6
∠4≅∠2
l∥n
Here is the correct order of the justifications to prove the Alternate Interior Angles Theorem:
1) ∠2 ≅ ∠6 (Corresponding Angles Postulate, because angles 2 and 6 are corresponding angles formed by the transversal line)
2) ∠4 ≅ ∠2 (Vertical Angles Theorem, because angles 4 and 2 are vertical angles)
3) ∠6 ≅ ∠4 (Symmetric Property of Congruence, because if ∠2 ≅ ∠6 and ∠4 ≅ ∠2, then ∠6 ≅ ∠4)
4) l ∥ n (Given, the lines l and n are parallel)
To prove the Alternate Interior Angles Theorem, we need to establish that the alternate interior angles formed by the two parallel lines are congruent. Here's how we can justify each step in the proof:
Step 1: ∠2 ≅ ∠6
- Justification: Vertical angles are congruent.
Step 2: ∠4 ≅ ∠2
- Justification: If we consider the transversal line crossing the parallel lines, ∠4 and ∠2 are corresponding angles, and corresponding angles formed by a transversal and parallel lines are congruent.
Step 3: ∠6 ≅ ∠4
- Justification: By using the Transitive Property of Congruence, we can conclude that if ∠2 ≅ ∠6 and ∠4 ≅ ∠2, then ∠6 ≅ ∠4.
Step 4: l ∥ n
- Justification: Given that lines l and n are parallel.
So, to complete the proof of the Alternate Interior Angles Theorem, we need to match the justifications for each step as follows:
Step 1: ∠2 ≅ ∠6 (Vertical angles are congruent)
Step 2: ∠4 ≅ ∠2 (Corresponding angles of transversal and parallel lines)
Step 3: ∠6 ≅ ∠4 (Transitive Property of Congruence)
Step 4: l ∥ n (Given that lines l and n are parallel)