select the correct answer from each drop - down menu. 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 |
| --- | --- |
| 1 ( ce = cd + de ); ( df = ef + de ) | 1 segment addition |
| 2 ( cd = ef ) | 2 given |
| 3 ( df = cd + de ) | 3 substitution property of equality |
| 4 (square) | 4 (square) |
| 5 ( ab = ce ) | 5 given |
| 6 ( ab = df ) | 6 transitive property of equality |

Question

Accepted Answer

# Explanation:
## Step1: Analyze Line 4's Requirement
We need to find the reason for the statement in line 4. The statement in line 5 is $ AB = CE $ (given), and line 6 is $ AB = DF $. Line 4 should connect $ CE $ and $ DF $ using the earlier steps. From step 2, $ CD = EF $, step 3: $ DF = CD + DE $, and step 1: $ CE = CD + DE $. So $ CE = DF $ (transitive property? Wait, no, earlier we have $ CE = CD + DE $ and $ DF = CD + DE $, so $ CE = DF $ by substitution (since $ CD + DE $ is common). Then line 4's statement should be $ CE = DF $ (to link with $ AB = CE $ and $ AB = DF $). Wait, the question is about the reason for line 4. Wait, the table:

Statements:
1. $ CE = CD + DE $; $ DF = EF + DE $ (segment addition)
2. $ CD = EF $ (given)
3. $ DF = CD + DE $ (substitution, since $ CD = EF $, so $ DF = CD + DE $)
4. [Blank]
5. $ AB = CE $ (given)
6. $ AB = DF $ (transitive, since $ AB = CE $ and $ CE = DF $)

Wait, step 3 is $ DF = CD + DE $, step 1 is $ CE = CD + DE $, so step 4 should be $ CE = DF $ (because $ CE = CD + DE $ and $ DF = CD + DE $, so by substitution or transitive? Wait, the reason for step 4: since $ CE = CD + DE $ (step 1) and $ DF = CD + DE $ (step 3), then $ CE = DF $ by the transitive property of equality (if $ a = b $ and $ b = c $, then $ a = c $; here $ a = CE $, $ b = CD + DE $, $ c = DF $). But also, since $ CD + DE $ is equal to both $ CE $ and $ DF $, so $ CE = DF $ by substitution (replacing $ CD + DE $ in $ CE = CD + DE $ with $ DF $ from step 3: $ DF = CD + DE $). Wait, the reason for line 4 (statement $ CE = DF $) would be substitution property (since we substitute $ CD + DE $ with $ DF $ in $ CE = CD + DE $) or transitive. But looking at the options (even though not fully shown, but from the problem, line 4's reason: since step 1: $ CE = CD + DE $, step 3: $ DF = CD + DE $, so $ CE = DF $ because they both equal $ CD + DE $, so by the transitive property of equality (or substitution). But the key is, the statement in line 4 is $ CE = DF $, and the reason is that $ CE = CD + DE $ (step 1) and $ DF = CD + DE $ (step 3), so $ CE = DF $ by substitution (substituting $ CD + DE $ with $ DF $ in $ CE = CD + DE $) or transitive. Wait, the problem is to find the statement and reason for line 4. Wait, the statement in line 4 should be $ CE = DF $, and the reason is "substitution property of equality" or "transitive property". But looking at the table, step 3's reason is "substitution property of equality" (since $ CD = EF $, so $ DF = CD + DE $ instead of $ DF = EF + DE $). Then step 4: since $ CE = CD + DE $ (step 1) and $ DF = CD + DE $ (step 3), then $ CE = DF $ by the transitive property? Wait, no, substitution: if $ CE = x $ and $ DF = x $, then $ CE = DF $ by substitution (x is $ CD + DE $). So the statement for line 4 is $ CE = DF $, and the reason is "transitive property of equality" or "substitution". But maybe the correct statement for line 4 is $ CE = DF $, and the reason is "transitive property" or "substitution". Wait, the problem is asking "What statement and reason belong in line 4?".

So step 4:
Statement: $ CE = DF $ (because $ CE = CD + DE $ (step 1) and $ DF = CD + DE $ (step 3), so they are equal)
Reason: Transitive property of equality (or substitution, since we substitute $ CD + DE $ with $ DF $ in $ CE = CD + DE $)

But let's re-examine:

1. $ CE = CD + DE $; $ DF = EF + DE $ (segment addition postulate)
2. $ CD = EF $ (given)
3. $ DF = CD + DE $ (substitution, because $ CD = EF $, so replace $ EF $ with $ CD $ in $ DF = EF + DE $)
4. $ CE = DF $ (because $ CE = CD + DE $ (step 1) and $ DF = CD + DE $ (step 3), so by substitution (replace $ CD + DE $ in $ CE = CD + DE $ with $ DF $) or transitive (if $ a = b $ and $ b = c $, then $ a = c $; here $ a = CE $, $ b = CD + DE $, $ c = DF $)
5. $ AB = CE $ (given)
6. $ AB = DF $ (transitive, since $ AB = CE $ and $ CE = DF $)

So the statement for line 4 is $ CE = DF $, and the reason is "substitution property of equality" (since we substitute $ CD + DE $ with $ DF $ in $ CE = CD + DE $) or "transitive property". But looking at the options, the reason for line 4 is likely "substitution" or "transitive". Wait, the problem's table:

Reasons:
1. segment addition
2. given
3. substitution property of equality
4. [Blank]
5. given
6. transitive property of equality

So line 4's reason: since $ CE = CD + DE $ (step 1) and $ DF = CD + DE $ (step 3), then $ CE = DF $ by substitution (because $ CD + DE $ is equal to $ DF $, so substitute $ DF $ for $ CD + DE $ in $ CE = CD + DE $), so the reason is "substitution property of equality"? Wait, no, step 3 is $ DF = CD + DE $ (substitution from step 2: $ CD = EF $, so $ DF = CD + DE $). Then step 4: $ CE = DF $ because $ CE = CD + DE $ (step 1) and $ DF = CD + DE $ (step 3), so by the transitive property (if $ CE = CD + DE $ and $ DF = CD + DE $, then $ CE = DF $). But the transitive property is used in step 6. Wait, maybe the statement in line 4 is $ CE = DF $, and the reason is "transitive property" or "substitution". Alternatively, maybe the statement in line 4 is $ CE = DF $, and the reason is "substitution" because we substitute $ CD + DE $ with $ DF $ (from step 3) into $ CE = CD + DE $ (step 1).

So the correct statement for line 4 is $ CE = DF $, and the reason is "substitution property of equality" (or transitive). But based on the table, step 3 is substitution, step 6 is transitive. So line 4's reason is likely "transitive" or "substitution". Wait, the problem is to select the correct answer for line 4. The statement should be $ CE = DF $, and the reason is "transitive property of equality" (since $ CE = CD + DE $ and $ DF = CD + DE $, so $ CE = DF $ by transitive) or "substitution". But let's check the flow:

- Step 1: $ CE = CD + DE $; $ DF = EF + DE $ (segment addition)
- Step 2: $ CD = EF $ (given)
- Step 3: $ DF = CD + DE $ (substitution, replace $ EF $ with $ CD $ in $ DF = EF + DE $)
- Step 4: $ CE = DF $ (because $ CE = CD + DE $ (step 1) and $ DF = CD + DE $ (step 3), so by substitution (replace $ CD + DE $ with $ DF $ in $ CE = CD + DE $) → reason: substitution property
- Step 5: $ AB = CE $ (given)
- Step 6: $ AB = DF $ (transitive, since $ AB = CE $ and $ CE = DF $)

Yes, so step 4's statement is $ CE = DF $, and the reason is "substitution property of equality" (because we substitute $ DF $ (which equals $ CD + DE $) into $ CE = CD + DE $ to get $ CE = DF $).

# Answer:
Statement: $ CE = DF $; Reason: Substitution Property of Equality

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Analyze Line 4's Requirement

Answer:

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