Question

given: cd = ef, ab = ce
prove: ab = df
the lengths ce = cd + de and df = ef + de by segment addition. it was given that cd = ef and applying the substitution property of equality gives df = cd + de. since both ce and df equal the same quantity, ce = df by the transitive property of equality. it was also given that ab = ce. applying the transitive property of equality again, ab = df.
use the paragraph proof to complete the two - column proof.
what statement and reason belong in line 4?

statements	reasons
2. cd = ef	2. given
3. df = cd + de	3. substitution property of equality
4. df = ce	4. ?
5. ab = ce	5. given
6. ab = df	6. transitive property of equality

reasons options: subtraction property of equality, transitive property of equality, given, addition property of equality|

Explanation:

Step1: Analyze the proof structure

We know from the given information and previous steps that \( CD = EF \) (given), \( CE = CD + DE \) (segment addition), \( DF = EF + DE \) (segment addition). We need to find the reason for \( DF = CE \) in line 4.

Step2: Recall properties of equality

Since \( CD = EF \) (given), we can substitute \( CD \) with \( EF \) in the equation \( CE = CD + DE \). So \( CE = EF + DE \), and we also have \( DF = EF + DE \) (from segment addition). By the transitive property of equality (if \( a = b \) and \( b = c \), then \( a = c \)), we can say \( DF = CE \) because both \( DF \) and \( CE \) equal \( EF + DE \). But looking at the reasons provided, the reason for line 4 (where \( DF = CE \)) should be the substitution property? Wait, no, let's re - check. Wait, we have \( CD = EF \) (given), \( CE = CD + DE \) (segment addition), \( DF = EF + DE \) (segment addition). Then, substitute \( CD \) with \( EF \) in the equation for \( CE \), we get \( CE = EF + DE \), and \( DF = EF + DE \), so by substitution (or transitive), \( DF = CE \). But the reason options include "substitution property of equality" for line 3, and for line 4, since we have \( DF = EF + DE \) and \( CE = EF + DE \) (after substitution), the reason for \( DF = CE \) is the transitive property? Wait, no, the problem is asking for the statement and reason in line 4. Wait, the statement in line 4 is \( DF = CE \). Let's track the steps:

1. \( CE = CD + DE \) (segment addition)
2. \( CD = EF \) (given)
3. \( DF = CD + DE \) (substitution: replace \( CD \) with \( EF \) in \( DF = EF + DE \)? Wait, no, \( DF = EF + DE \), and \( CD = EF \), so substitute \( CD \) for \( EF \) in \( DF = EF + DE \), we get \( DF = CD + DE \) (that's line 3, reason: substitution property of equality)
4. Now, from line 1, \( CE = CD + DE \), and from line 3, \( DF = CD + DE \). So by the transitive property of equality (if \( a = b \) and \( c = b \), then \( a = c \)), we can say \( DF = CE \). But looking at the reason options for line 4, the options are "DF = CE", and the reason options include "given", "subtraction property", "negative property", "addition property", "transitive property"? Wait, no, the reason for line 4 (statement \( DF = CE \)) is that both \( DF \) and \( CE \) are equal to \( CD + DE \) (from line 3 and line 1), so by the transitive property of equality. But maybe the problem is simpler. Wait, the key is that we have \( AB = CE \) (given) and we want to prove \( AB = DF \). But for line 4, the statement is \( DF = CE \), and the reason should be the transitive property? Wait, no, let's look at the given information again. We know \( AB = CE \) (given), \( CD = EF \) (given), \( CE = CD + DE \) (segment addition), \( DF = EF + DE \) (segment addition). Then:

- Substitute \( CD \) with \( EF \) in \( CE = CD + DE \), we get \( CE = EF + DE \)
- \( DF = EF + DE \) (segment addition)
- So \( CE = DF \) (transitive property: \( CE = EF + DE \) and \( DF = EF + DE \), so \( CE = DF \))
- Then, since \( AB = CE \) (given), by transitive property, \( AB = DF \)

But for line 4, the statement is \( DF = CE \), and the reason is the transitive property of equality? Wait, but the reason options for line 4 (from the image, the reason column for line 4 has options like "given", "subtraction property of equality", "negative property of equality", "addition property of equality", "transitive property"? Wait, no, the user's image shows the reason for line 4 as a dropdown with options: "DF = CE", and the reason options are "given", "subtraction property of equality", "negative property of equality", "addition property of equality", "transitive property"? Wait, no, the original problem's reason for line 4: let's re - examine the text. The problem says "What statement and reason belong in line 4?".

Wait, let's list the steps properly:

- Step 1: \( CE = CD + DE \) (Segment Addition Postulate)
- Step 2: \( CD = EF \) (Given)
- Step 3: \( DF = CD + DE \) (Substitution Property: substitute \( CD \) for \( EF \) in \( DF = EF + DE \))
- Step 4: \( DF = CE \) (Transitive Property of Equality: since \( CE = CD + DE \) (Step 1) and \( DF = CD + DE \) (Step 3), then \( DF = CE \))

But if we look at the reason options for line 4, and considering the flow, the reason for \( DF = CE \) is that both \( DF \) and \( CE \) are equal to \( CD + DE \), so by the transitive property. However, if we consider the given that \( AB = CE \) (given), and we can also think of it as \( DF = CE \) because of the substitution and transitive. But maybe the answer is that the statement in line 4 is \( DF = CE \) and the reason is the transitive property of equality (or substitution, but based on the steps, it's transitive). But since the problem is about segment lengths and equality properties, the key is that we use the fact that if two quantities are equal to the same quantity, they are equal to each other (transitive property).

Answer:

The statement in line 4 is \( \boldsymbol{DF = CE} \) and the reason is the transitive property of equality (or substitution, but based on the step - by - step, since \( CE = CD + DE \) and \( DF = CD + DE \), by transitive, \( DF = CE \)). If we follow the given reason options (from the image, the reason for line 4 is likely the transitive property, but if we consider the options provided in the small box, the reason for \( DF = CE \) is that both are equal to \( CD + DE \), so the reason is the transitive property of equality.