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Question 1179597: Formulate a simple geometric statement that can be proven using
any combination of the following: congruence axioms, substitution, and
definitions. Show two different ways to prove this statement.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Okay, let's formulate a simple geometric statement and prove it in two different ways using congruence axioms, substitution, and definitions.
**Geometric Statement:**
**Given:** Two line segments AB and CD, where AB ≅ CD. Let E be a point on AB such that AE ≅ EB. Let F be a point on CD such that CF ≅ FD.
**Prove:** AE ≅ CF.
**Proof 1: Using Substitution and Congruence**
| Statement | Reason |
|---|---|
| 1. AB ≅ CD | Given |
| 2. AE ≅ EB | Given |
| 3. CF ≅ FD | Given |
| 4. AE = EB | Definition of congruent segments (If segments are congruent, their lengths are equal) |
| 5. CF = FD | Definition of congruent segments |
| 6. AB = CD | Definition of congruent segments |
| 7. AE + EB = AB | Segment Addition Postulate |
| 8. CF + FD = CD | Segment Addition Postulate |
| 9. AE + AE = AB | Substitution (Substitute AE for EB in step 7) |
| 10. CF + CF = CD | Substitution (Substitute CF for FD in step 8) |
| 11. 2AE = AB | Simplification |
| 12. 2CF = CD | Simplification |
| 13. 2AE = CD | Substitution (Substitute CD for AB in step 11, using step 6) |
| 14. 2AE = 2CF | Substitution (Substitute 2CF for CD in step 13, using step 12) |
| 15. AE = CF | Division Property of Equality (Divide both sides by 2) |
| 16. AE ≅ CF | Definition of congruent segments |
**Proof 2: Using a Different Substitution Approach**
| Statement | Reason |
|---|---|
| 1. AB ≅ CD | Given |
| 2. AE ≅ EB | Given |
| 3. CF ≅ FD | Given |
| 4. AE = EB | Definition of congruent segments |
| 5. CF = FD | Definition of congruent segments |
| 6. AB = CD | Definition of congruent segments |
| 7. AE + EB = AB | Segment Addition Postulate |
| 8. CF + FD = CD | Segment Addition Postulate |
| 9. AB/2 = AE | Division Property of Equality and Substitution(Using step 4, 7, and the fact that since AE=EB, AB is 2AE) |
| 10. CD/2 = CF | Division Property of Equality and Substitution(Using step 5, 8, and the fact that since CF=FD, CD is 2CF) |
| 11. AB/2 = CD/2 | Division Property of Equality (Dividing both sides of step 6 by 2) |
| 12. AE = CD/2 | Substitution (Substitute AE for AB/2 in step 11, using step 9) |
| 13. AE = CF | Substitution(Substitute CF for CD/2 in step 12, using step 10)|
| 14. AE ≅ CF | Definition of congruent segments |
**Explanation of the Proofs:**
* Both proofs rely on the fundamental definitions of congruent segments, which state that if segments are congruent, their lengths are equal, and vice versa.
* The Segment Addition Postulate is used to express the lengths of the given segments in terms of their parts.
* The proofs then use substitution to manipulate the equations and arrive at the desired conclusion.
* The division property of equality is used to isolate the segment lengths.
These proofs are simple yet demonstrate the power of combining definitions, postulates, and substitution to prove geometric statements.
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