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Write the following paragraph proof as a two-column proof.

Given: [tex]A B = C D[/tex] and [tex]B C = D E[/tex]
Prove: [tex]A C = C E[/tex]

Statements | Reasons
---|---
1. [tex]A B = C D[/tex] | Given
2. [tex]B C = D E[/tex] | Given
3. [tex]A B + B C = C D + B C[/tex] | Addition Property of Equality
4. [tex]C D + B C = C D + D E[/tex] | Substitution Property of Equality (substitute [tex]B C[/tex] with [tex]D E[/tex])
5. [tex]A B + B C = A C[/tex] | Segment Addition Postulate
6. [tex]C D + D E = C E[/tex] | Segment Addition Postulate
7. [tex]A C = C E[/tex] | Substitution Property of Equality (substitute [tex]A B + B C[/tex] with [tex]A C[/tex] and [tex]C D + D E[/tex] with [tex]C E[/tex])

Type the correct answer in the box.
B [tex]I \underline{\cup} X^2 X_2 14 p t[/tex]

Sagot :

GinnyAnswer
Here's a detailed two-column proof for the given problem.

| Statements | Reasons |
|-----------------------------------------------|----------------------------------------------|
| 1. [tex]\( AB = CD \)[/tex] | Given |
| 2. [tex]\( BC = DE \)[/tex] | Given |
| 3. [tex]\( AB + BC = CD + BC \)[/tex] | Addition property of equality |
| 4. [tex]\( CD + BC = CD + DE \)[/tex] | Substitution property of equality |
| 5. [tex]\( AB + BC = AC \)[/tex] | Segment addition (AB + BC = AC) |
| 6. [tex]\( CD + DE = CE \)[/tex] | Segment addition (CD + DE = CE) |
| 7. [tex]\( AC = CE \)[/tex] | Substitution property of equality |

Let's break down each part of the proof:

1. Statement: [tex]\( AB = CD \)[/tex]
- Reason: Given

2. Statement: [tex]\( BC = DE \)[/tex]
- Reason: Given

3. Statement: [tex]\( AB + BC = CD + BC \)[/tex]
- Reason: Addition property of equality (we add [tex]\( BC \)[/tex] to both sides of the equalities in previous steps)

4. Statement: [tex]\( CD + BC = CD + DE \)[/tex]
- Reason: Substitution property of equality (replacing [tex]\( BC \)[/tex] with [tex]\( DE \)[/tex] based on step 2)

5. Statement: [tex]\( AB + BC = AC \)[/tex]
- Reason: Segment addition (defining [tex]\( AC \)[/tex] as the combined length of segments [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex])

6. Statement: [tex]\( CD + DE = CE \)[/tex]
- Reason: Segment addition (defining [tex]\( CE \)[/tex] as the combined length of segments [tex]\( CD \)[/tex] and [tex]\( DE \)[/tex])

7. Statement: [tex]\( AC = CE \)[/tex]
- Reason: Substitution property of equality (replacing [tex]\( AB + BC \)[/tex] with [tex]\( AC \)[/tex] and [tex]\( CD + DE \)[/tex] with [tex]\( CE \)[/tex] from steps 5 and 6)
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