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 = ef reason: transitive property of equality

Explanation:

Step1: Analyze given equal - segments

We know \(AB = CE\) and \(CD=EF\). To prove \(AB = DF\), we need to manipulate the segment - length equalities.

Step2: Consider segment addition

We start from \(CD = EF\). By the addition property of equality, if we add \(DE\) to both sides of the equation \(CD = EF\), we get \(CD + DE=DE + EF\). Since \(CE=CD + DE\) and \(DF=DE + EF\) (by segment addition postulate), we can say \(CE = DF\).

Answer:

C. Statement: \(CE = DF\)
Reason: Transitive Property of Equality (although the more direct reason from the steps above is segment - addition and substitution, but among the given options, this is the most logical next step as it helps in the chain of equalities to reach \(AB = DF\) since \(AB = CE\))