Sage has already proved that ∆IGF ~ ∆FGH. She is now attempting to prove that IJ ll FH. Help her to construct a viable argument. What is the missing statement and reason in her proof?

Question

Statements
1. ∆IGJ ~ ∆FGH
2.
3. IJ ll FH

Statements
1. Given
2.
3. If corresponding angles are congruent, then lines are parallel

a. The missing statement is ∠JIF ≅ ∠GFH and the missing reason is "corresponding angles of similar triangles are congruent"
b. The missing statement is ∠GJI ≅ ∠GHF and the missing reason is "corresponding angles of similar triangles are congruent"
c. The missing statement is ∠GIJ ≅ ∠GFH and the missing reason is "CPCTC Theorem"
d. The missing statement is ∠JIF ≅ ∠GFH and the missing reason is "CPCTC Theorem"

Answer 1

To prove that line segment IJ is parallel to line segment FH, we need to focus on finding corresponding angles between the two similar triangles ∆IGF and ∆FGH. Because Sage has already established that ∆IGF ~ ∆FGH, we know that corresponding angles of these triangles are congruent.

Therefore, the correct missing statement would be that the angles that correspond between the triangles are congruent, which leads us to the conclusion that IJ || FH according to the property that if two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

Looking at your options, the best choice is:

a. The missing statement is ∠JIF ≅ ∠GFH and the missing reason is "corresponding angles of similar triangles are congruent."

This provides both the statement needed for the proof and the appropriate reason to conclude that IJ is parallel to FH.