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Logic and Proof

Logic and Proof

At a Glance - Proofs


When detectives Benson and Stabler search a suspect's apartment, they aren't looking for just anything to give to the prosecutor. They need solid, tangible, and legally obtained evidence. (Although "legally obtained" is more of a suggestion than a requirement on Law and Order.) The point is that no jury or judge will convict based on a hunch or a guess. They'll need proof.

Think of proofs as arguments. If you're debating with your parents over your curfew and your main point is that you're responsible enough to stay out late, it's important to have reasons supporting this claim. Just an FYI, now wouldn't be a good time to bring up that flashing red light you ran a couple sections back.

The ability to support your statements with reasons is the essence of effective debate, and it's probably the most important skill to gain from studying geometry (and mathematics in general). It'll help you to be a more convincing speaker and might just get your curfew pushed back an hour or five.

Example 1

What is the difference between a postulate and a theorem?


Example 2

Complete the following proof.

Given: A = B, C = D
Prove: X(A + C) = BX + DX


Example 3

Complete the following proof.

Given: A = B, C = D, C ≠ 0
Prove:


Exercise 1

What is the reason for statement 7?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = C?

Exercise 2

What is the reason for statement 8?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0?

Exercise 3

What is the reason for statement 9?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0 

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BB?

Exercise 4

What is statement 11 given its reason?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BBMultiplication Property (1)
10. AB = CBSubstitution Property (9 and 2)
11. ?Division Property (10 and 6)

Exercise 5

What is statement 12 given its reason?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BBMultiplication Property (1)
10. AB = CBSubstitution Property (9 and 2)
11. Division Property (10 and 6)
12. ?Substitution Property (11 and 5)

Exercise 6

What is statement 13 given its reason?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove: 

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BBMultiplication Property (1)
10. AB = CBSubstitution Property (9 and 2)
11. Division Property (10 and 6)
12.Substitution Property (11 and 5)
13. ?Multiplication Property (12)

Exercise 7

What is the reason for statement 14?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove:

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BBMultiplication Property (1)
10. AB = CBSubstitution Property (9 and 2)
11. Division Property (10 and 6)
12. Substitution Property (11 and 5)
13. Multiplication Property (12)
14. X = Z?

Exercise 8

What is the reason for statement 15?

Given: A = B, B = C, X = Y, Y = Z, H = G, G ≠ 0

Prove:

StatementsReasons
1. A = BGiven
2. B = CGiven
3. X = YGiven
4. Y = ZGiven
5. H = GGiven
6. G ≠ 0Given
7. A = CTransitive Property (1 and 2)
8. H ≠ 0Substitution Property (5 and 6)
9. AB = BBMultiplication Property (1)
10. AB = CBSubstitution Property (9 and 2)
11. Division Property (10 and 6)
12. Substitution Property (11 and 5)
13. Multiplication Property (12)
14. X = ZTransitive Property (3 and 4)
15. ?

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