Answer and Step-by-step explanation:
To prove that AC = CE given AB = CD and BC = DE, we can use a two-column proof. Here's how we can demonstrate this:
Statement | Reason
-----------------------------------|-----------------------------------------
1. AB = CD | Given
2. BC = DE | Given
3. AB + BC = CD + DE | Addition property of equality (adding equal quantities to both sides)
4. AC = CD | Definition of a line segment (AB + BC = AC)
5. CD + DE = AC + CE | Transitive property of equality (substituting CD for AC)
6. AC + CE = AC + CE | Reflexive property of equality (equality of a quantity to itself)
7. CD + DE = AC + CE | Transitive property of equality (substituting AC + CE for CD + DE)
8. AC = CE | Subtraction property of equality (subtracting equal quantities from both sides)
By using the properties of equality and the given information, we have shown that AC is equal to CE.