Question

select the correct answer from each drop - down menu.
in the figure, ( cd = ef ) and ( ab = ce ). complete the statements to prove that ( ab = df ).
( cd + de = ef + de ) by the property of equality.
( ce = cd + de ) and ( df = ef + de ) by.
( ce = df ) by the property of equality.
given, ( ab = ce ) and ( ce = df ) implies ( ab = df ) by the property of equality.

Explanation:

Step1: Addition Property of Equality

If \(a = b\), then \(a + c=b + c\). Here \(CD = EF\), adding \(DE\) to both sides gives \(CD + DE=EF + DE\) by the Addition Property of Equality.

Step2: Segment Addition Postulate

The Segment Addition Postulate states that if \(D\) is between \(C\) and \(E\), then \(CE=CD + DE\) and if \(E\) is between \(D\) and \(F\), then \(DF = EF+DE\)

Step3: Substitution Property of Equality

Since \(CD + DE=EF + DE\), and \(CE = CD + DE\), \(DF=EF + DE\), we substitute \(CD + DE\) with \(CE\) and \(EF + DE\) with \(DF\) (because \(CD = EF\)) to get \(CE = DF\) by the Substitution Property of Equality (\(a=b\), \(b = c\) implies \(a=c\) in the context of substitution here as \(CD + DE\) is substituted with \(CE\) and \(EF + DE\) with \(DF\) given \(CD = EF\))

Step4: Transitive Property of Equality

The Transitive Property of Equality states that if \(a=b\) and \(b = c\), then \(a=c\). Given \(AB = CE\) and \(CE=DF\), we have \(AB = DF\) by the Transitive Property of Equality

Answer:

1. Addition
2. Segment Addition Postulate
3. Substitution
4. Transitive