From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!

Proofs Exercises

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. ?