Example 1
What is the reason for statement 7?
Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0
Prove: 
| Statements | Reasons |
| 1. A = B | Given |
| 2. B = C | Given |
| 3. X = Y | Given |
| 4. Y = Z | Given |
| 5. H = G | Given |
| 6. G ≠ 0 | Given |
| 7. A = C | ? |
Example 2
What is the reason for statement 8?
Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0
Prove:
| Statements | Reasons |
| 1. A = B | Given |
| 2. B = C | Given |
| 3. X = Y | Given |
| 4. Y = Z | Given |
| 5. H = G | Given |
| 6. G ≠ 0 | Given |
| 7. A = C | Transitive Property (1 and 2) |
| 8. H ≠ 0 | ? |
Example 3
What is the reason for statement 9?
Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0
Prove:
| Statements | Reasons |
| 1. A = B | Given |
| 2. B = C | Given |
| 3. X = Y | Given |
| 4. Y = Z | Given |
| 5. H = G | Given |
| 6. G ≠ 0 | Given |
| 7. A = C | Transitive Property (1 and 2) |
| 8. H ≠ 0 | Substitution Property (5 and 6) |
| 9. AB = BB | ? |
Example 4
What is statement 11 given its reason?
Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0
Prove:
| Statements | Reasons |
| 1. A = B | Given |
| 2. B = C | Given |
| 3. X = Y | Given |
| 4. Y = Z | Given |
| 5. H = G | Given |
| 6. G ≠ 0 | Given |
| 7. A = C | Transitive Property (1 and 2) |
| 8. H ≠ 0 | Substitution Property (5 and 6) |
| 9. AB = BB | Multiplication Property (1) |
| 10. AB = CB | Substitution Property (9 and 2) |
| 11. ? | Division Property (10 and 6) |
Example 5
What is statement 12 given its reason?
Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0
Prove:
| Statements | Reasons |
| 1. A = B | Given |
| 2. B = C | Given |
| 3. X = Y | Given |
| 4. Y = Z | Given |
| 5. H = G | Given |
| 6. G ≠ 0 | Given |
| 7. A = C | Transitive Property (1 and 2) |
| 8. H ≠ 0 | Substitution Property (5 and 6) |
| 9. AB = BB | Multiplication Property (1) |
| 10. AB = CB | Substitution Property (9 and 2) |
11.  | Division Property (10 and 6) |
| 12. ? | Substitution Property (11 and 5) |
Example 6
What is statement 13 given its reason?
Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0
Prove:
| Statements | Reasons |
| 1. A = B | Given |
| 2. B = C | Given |
| 3. X = Y | Given |
| 4. Y = Z | Given |
| 5. H = G | Given |
| 6. G ≠ 0 | Given |
| 7. A = C | Transitive Property (1 and 2) |
| 8. H ≠ 0 | Substitution Property (5 and 6) |
| 9. AB = BB | Multiplication Property (1) |
| 10. AB = CB | Substitution Property (9 and 2) |
| 11. | Division Property (10 and 6) |
12. | Substitution Property (11 and 5) |
| 13. ? | Multiplication Property (12) |
Example 7
What is the reason for statement 14?
Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0
Prove:
| Statements | Reasons |
| 1. A = B | Given |
| 2. B = C | Given |
| 3. X = Y | Given |
| 4. Y = Z | Given |
| 5. H = G | Given |
| 6. G ≠ 0 | Given |
| 7. A = C | Transitive Property (1 and 2) |
| 8. H ≠ 0 | Substitution Property (5 and 6) |
| 9. AB = BB | Multiplication Property (1) |
| 10. AB = CB | Substitution Property (9 and 2) |
| 11. | Division Property (10 and 6) |
| 12. | Substitution Property (11 and 5) |
13.  | Multiplication Property (12) |
| 14. X = Z | ? |
Example 8
What is the reason for statement 15?
Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0
Prove:
| Statements | Reasons |
| 1. A = B | Given |
| 2. B = C | Given |
| 3. X = Y | Given |
| 4. Y = Z | Given |
| 5. H = G | Given |
| 6. G ≠ 0 | Given |
| 7. A = C | Transitive Property (1 and 2) |
| 8. H ≠ 0 | Substitution Property (5 and 6) |
| 9. AB = BB | Multiplication Property (1) |
| 10. AB = CB | Substitution Property (9 and 2) |
| 11. | Division Property (10 and 6) |
| 12. | Substitution Property (11 and 5) |
| 13. | Multiplication Property (12) |
| 14. X = Z | Transitive Property (3 and 4) |
| 15. | ? |