Question

given: ab = ce
cd = ef
prove: ab = df
what is the next step in the given proof? choose the most logical approach
a. statement: ab = df
reason: substitution property of equality
b. statement: de = de
reason: subtraction property of equality
c. statement: ce = df
reason: transitive property of equality
d. statement: cd = de
reason: substitution property of equality
e. statement: ce = de + ef
reason: transitive property of equality

Explanation:

Step1: Analyze each option

• Option A: We need to prove \(AB = DF\), so directly stating \(AB = DF\) as the next step without proper substitution based on previous valid steps is incorrect.
• Option B: \(DE = DE\) is a reflexive property (a quantity is equal to itself), but the subtraction property of equality is not relevant here. The previous step was using the addition property of equality (\(CD+DE=DE + EF\)) and we know \(CE=CD + DE\) and \(DF=DE+EF\) from segment addition.
• Option C: Since \(CE = CD + DE\) and \(DF=DE + EF\), and \(CD = EF\) (given), by substitution \(CE=DF\). Then, because \(AB = CE\) (given), using the transitive property of equality (\(a=b\) and \(b = c\) implies \(a=c\)), we can eventually get \(AB=DF\). This is a logical next - step after establishing \(CE = DF\).
• Option D: There is no basis to state \(CD = DE\) as \(CD = EF\) is given and there is no information about \(DE\) being equal to \(CD\) from the given data.
• Option E: \(CE=CD + DE\) is from segment addition, not the transitive property of equality. The transitive property is for equality of three quantities (\(a=b\), \(b = c\) then \(a=c\)).

Answer:

C. Statement: \(CE = DF\) Reason: Transitive Property of Equality