Question

complete the proof that $\triangle ghi \cong \triangle fej$.
(image of two triangles with markings and a table of statements and reasons:

1. $\angle ghi \cong \angle fej$ (given)
2. $\angle gih \cong \angle ejf$ (given)
3. $\overline{fg} \cong \overline{ij}$ (given)
4. $fj = ij + fi$ (additive property of length)
5. $gi = fg + fi$ (reason blank)
6. $fj = fg + fi$ (reason blank)
7. $gi = fj$ (reason blank)
8. $\triangle ghi \cong \triangle fej$ (aas))

Explanation:

Step1: Analyze Statement 5

To find the reason for \( GI = FG + FI \), we look at the segment addition. From the diagram, \( GI \) is composed of \( FG \) and \( FI \) (since \( F \) is on \( GI \)? Wait, no, looking at the diagram, \( FG \cong IJ \) (statement 3), and \( FJ = IJ + FI \) (statement 4). Wait, actually, the segment \( GI \) can be seen as \( FG + FI \) by the Additive Property of Length, similar to statement 4. So the reason for \( GI = FG + FI \) is the Additive Property of Length, as it's the sum of two segments making up \( GI \).

Step2: Analyze Statement 6

For \( FJ = FG + FI \), we know from statement 3 that \( FG \cong IJ \), so \( FG = IJ \) (congruent segments have equal length). Then from statement 4, \( FJ = IJ + FI \), so substituting \( IJ \) with \( FG \) (since \( FG = IJ \)) gives \( FJ = FG + FI \). So the reason is Substitution Property (using \( FG = IJ \) from statement 3 and statement 4).

Step3: Analyze Statement 7

For \( GI = FJ \), from statement 5 (\( GI = FG + FI \)) and statement 6 (\( FJ = FG + FI \)), by the Transitive Property of Equality (if \( a = b \) and \( b = c \), then \( a = c \)), we get \( GI = FJ \).

But focusing on statement 5 first: the reason for \( GI = FG + FI \) is the Additive Property of Length, same as statement 4 but for segment \( GI \).

Answer:

For statement 5, the reason is "Additive Property of Length"; for statement 6, the reason is "Substitution (using \( \overline{FG} \cong \overline{IJ} \) to replace \( IJ \) with \( FG \) in \( FJ = IJ + FI \))"; for statement 7, the reason is "Transitive Property of Equality". But if we are to fill the first blank (statement 5's reason), it's "Additive Property of Length".