Question

4 write a justification for each step. given that ( eg = fh ) diagram: e, f, g, h on a line segment (\begin{array}{|l|l|} hline \text{statement} & \text{reason} \\ hline 1. eg = fh & 1. \text{given information} \\ hline 2. eg = ef + fg; fh = fg + gh & 2. \text{a} \\ hline 3. ef + fg = fg + gh & 3. \text{b} \\ hline 4. ef = gh & 4. \text{subtraction property of equality} \\ hline end{array}) options: (\boldsymbol{\text{a}}) a segment addition postulate b substitution property of equality (\boldsymbol{\text{b}}) a segment addition postulate b definition of congruent segments (\boldsymbol{\text{c}}) a angle addition postulate b subtraction property of equality (\boldsymbol{\text{d}}) a substitution property of equality b transitive property of equality

Explanation:

• For [A]: The Segment Addition Postulate states that if a point \( F \) is between \( E \) and \( G \), then \( EG = EF + FG \), and similarly for \( FH = FG + GH \) (since \( G \) is between \( F \) and \( H \)). So [A] should be the Segment Addition Postulate.
• For [B]: We know \( EG = FH \) (given) and \( EG = EF + FG \), \( FH = FG + GH \) (from [A]). Substituting \( EG \) and \( FH \) in the equation \( EG = FH \) gives \( EF + FG = FG + GH \), which is the Substitution Property of Equality (replacing equal quantities with each other). The other options for [B] (Definition of Congruent Segments, Subtraction Property, Transitive Property) don't fit: Definition of Congruent Segments is about segment congruence (not equality substitution here), Subtraction Property is used later in step 4, and Transitive Property is \( a = b, b = c \implies a = c \), not applicable here. So the correct pair is [A] Segment Addition Postulate and [B] Substitution Property of Equality, which is option A.

Answer:

A. [A] Segment Addition Postulate; [B] Substitution Property of Equality