Question

b. write a two - column proof.
statements\treasons
\tgiven
\tdefinition of congruent segments
\tgiven
\tdefinition of congruent segments
\tsegment addition postulate
\tsegment addition postulate
\tsubstitution property of equality
\tsubstitution property of equality
\tdefinition of congruent segments
boxed statements: ( rm = cd ), ( rs = cf ), ( sm = fd ), ( rm cong cd ), ( rs cong cf ), ( sm cong fd ), ( cd = cf + fd ), ( rm = rs + sm ), ( rs + sm = cd )

Explanation:

To solve this two - column proof problem, we will follow the logical flow of statements and their corresponding reasons. Let's assume we are trying to prove a relationship between segments (for example, maybe \(RM\cong CD\) or some other congruence/equality). Here is a step - by - step construction of the two - column proof:

Step 1: Start with the given information

Statements	Reasons
\(SM = FD\)	Given
\(RM=RS + SM\)	Segment Addition Postulate (The whole segment \(RM\) is the sum of its parts \(RS\) and \(SM\))
\(CD = CF+FD\)	Segment Addition Postulate (The whole segment \(CD\) is the sum of its parts \(CF\) and \(FD\))

Step 2: Use the Substitution Property of Equality

Statements	Reasons

Step 3: Use the Substitution Property of Equality again

Statements	Reasons

Step 4: Use the Definition of Congruent Segments

Statements	Reasons

(Note: The exact proof may vary depending on the specific goal of the proof, but this is a general structure based on the given segments and reasons. If the goal was to prove \(RS + SM=CD\) or some other related statement, the steps would be adjusted accordingly, but the use of segment addition postulate, substitution property, and definition of congruent segments remains consistent.)

If we assume the final answer is to prove \(RM\cong CD\), then:

Answer:

\(RM\cong CD\) (proven through the two - column proof as above)