Question

4 write a justification for each step. given that eg = fh.
statement\treason

1. eg = fh\t1. given information
2. eg = ef + fg; fh = fg + gh\t2. a
3. ef + fg = fg + gh\t3. substitution
4. ef = gh\t4. b

a a segment addition postulate b subtraction property of equality
b a addition property of equality b transitive property of equality
c a segment addition postulate b definition of congruent segments
d a segment addition postulate b substitution property of equality

Explanation:

Step1: Identify the rule for segment - length breakdown

The Segment Addition Postulate states that if we have three collinear points (in this case for points $E$, $F$, $G$ and $F$, $G$, $H$), the length of the whole segment is the sum of the lengths of its non - overlapping sub - segments. So, for segment $EG$ with points $E$, $F$, $G$ and segment $FH$ with points $F$, $G$, $H$, we have $EG = EF+FG$ and $FH=FG + GH$.

Step2: Apply substitution

Since $EG = FH$ (given) and $EG = EF + FG$, $FH=FG + GH$, we can substitute the expressions for $EG$ and $FH$ into the given equation, resulting in $EF + FG=FG + GH$.

Step3: Use the subtraction property of equality

The subtraction property of equality states that if $a + c=b + c$, then $a=b$. Here, $a = EF$, $b = GH$, and $c = FG$. Subtracting $FG$ from both sides of the equation $EF + FG=FG + GH$ gives us $EF=GH$.

Answer:

1. A [A] Segment Addition Postulate
2. B [B] Subtraction Property of Equality