Question

complete the proof by choosing the correct
eason\.
given: ( dg = fh ), ( eg = gh )
prove: ( de = fg )
diagram: horizontal line ( h-g-f ), vertical line ( g-e-d ) (downward from ( g ) to ( d ), with ( e ) between ( g ) and ( d ))

statement	reason
2 ( fh = fg + gh )	reason ?
3 ( dg = fh )	reason ?
4 ( de + eg = fg + gh )	reason ?
5 ( eg = gh )	reason ?
6 ( de + gh = fg + gh )	reason ?
7 ( de = fg )	reason ?

Explanation:

Step1: Analyze Statement 1

The statement \( DG = DE + EG \) is based on the Segment Addition Postulate, which states that if a point \( E \) is on segment \( DG \), then \( DG = DE + EG \).

Step2: Analyze Statement 2

The statement \( FH = FG + GH \) is also based on the Segment Addition Postulate, as point \( G \) is on segment \( FH \), so \( FH = FG + GH \).

Step3: Analyze Statement 3

The statement \( DG = FH \) is given in the problem, so the reason is "Given".

Step4: Analyze Statement 4

Since \( DG = DE + EG \) (from Step 1) and \( FH = FG + GH \) (from Step 2) and \( DG = FH \) (from Step 3), we can substitute \( DG \) with \( FH \) (or vice versa) using the Substitution Property of Equality. So \( DE + EG = FG + GH \) by Substitution.

Step5: Analyze Statement 5

The statement \( EG = GH \) is given in the problem, so the reason is "Given".

Step6: Analyze Statement 6

Since \( EG = GH \) (from Step 5), we can substitute \( EG \) with \( GH \) in the equation \( DE + EG = FG + GH \) (from Step 4) using the Substitution Property of Equality. So \( DE + GH = FG + GH \) by Substitution.

Step7: Analyze Statement 7

To get \( DE = FG \) from \( DE + GH = FG + GH \), we use the Subtraction Property of Equality, which states that if \( a + c = b + c \), then \( a = b \) (subtracting \( c \) from both sides). So we subtract \( GH \) from both sides of \( DE + GH = FG + GH \) to get \( DE = FG \).

Answer:

s for each Reason:

1. Segment Addition Postulate
2. Segment Addition Postulate
3. Given
4. Substitution Property of Equality (from Steps 1, 2, 3)
5. Given
6. Substitution Property of Equality (from Steps 4, 5)
7. Subtraction Property of Equality (subtract \( GH \) from both sides)