Sage has already proved that triangle IGJ ~ triangle FGH. she is now attempting to prove that line IJ || line FH. help her to construct a viable argument. what is the missing statement and reason in her proof.

Answer 1

To demonstrate that lines \( IJ \) and \( FH \) are parallel, given that triangles \( IGJ \) and \( FGH \) are similar (denoted by \( \triangle IGJ \sim \triangle FGH \)), we can utilize the properties of similar triangles.

Here’s how you can structure the argument, including the missing statement and reason:

1. Given: Triangles \( IGJ \) and \( FGH \) are similar.

2. Statement: Corresponding angles of similar triangles are congruent.

3. Reason: If \( \triangle IGJ \sim \triangle FGH \), then \(\angle IGJ \cong \angle FGH\) and \(\angle GJI \cong \angle GHF\).

4. Statement: If two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel.

5. Reason: This is the Converse of the Corresponding Angles Postulate.

6. Conclusion: Since \( \angle IGJ \cong \angle FGH \) and \( \angle GJI \cong \angle GHF \), it follows that \( IJ || FH \).

The Missing Piece in the Proof

The specific missing statement and reason from the original argument you were constructing might be:

• Missing Statement: \( \angle IGJ \cong \angle FGH \) or \( \angle GJI \cong \angle GHF \).

• Missing Reason: Corresponding angles in similar triangles are equal.

By incorporating this statement and reason into Sage's proof, she would have a complete argument establishing that \( IJ \parallel FH \).