Question

$$\overleftrightarrow{gh}\parallel\overleftrightarrow{fi}$$ and $$\angle g\cong\angle ifj$$. complete the proof that $$\angle i\cong\angle ghj$$.

statement	reason
2	$$\angle g\cong\angle ifj$$	given
3	$$\angle ifj\cong\angle ghj$$	alternate interior angles theorem
4	$$\angle g\cong\angle i$$
5	$$\angle ifj\cong\angle i$$	transitive property of congruence
6	$$\angle i\cong\angle ghj$$	transitive property of congruence

Explanation:

Step1: Identify the reason for ∠G ≅ ∠I

Since \(\overleftrightarrow{GH}\parallel\overleftrightarrow{FI}\), and \(FG\) is a transversal. By the Alternate Interior Angles Theorem, when two parallel lines are cut by a transversal, alternate - interior angles are congruent. So, the reason for \(\angle G\cong\angle I\) is the Alternate Interior Angles Theorem.

Step2: Identify the reason for \(\angle IFJ\cong\angle I\)

We know that \(\angle G\cong\angle IFJ\) (given) and \(\angle G\cong\angle I\) (from step 1). By the Transitive Property of Congruence (if \(a = b\) and \(a = c\), then \(b = c\)), we have \(\angle IFJ\cong\angle I\).

Answer:

The reason for statement 4 (\(\angle G\cong\angle I\)) is "Alternate Interior Angles Theorem" and the reason for statement 5 (\(\angle IFJ\cong\angle I\)) is "Transitive Property of Congruence".