# Logic and Proof

### 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 = BC | 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 = BC | 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 = BC | 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 = BC | 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 = BC | 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. | ? |