Mathematics 8 worksheet focusing on logical statements and their inverse, converse, and contrapositive forms.

Let's solve each problem on the Mathematics 8 Worksheet by identifying the inverse, converse, and contrapositive of each conditional statement.

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🔹 Definitions:

For a conditional statement:
If P, then Q
- Inverse: If not P, then not Q
- Converse: If Q, then P
- Contrapositive: If not Q, then not P

> Note: The contrapositive is logically equivalent to the original statement. The inverse and converse are not necessarily true even if the original is.

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Now let’s go through each question.

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1. If you visited Dallas, then you've been to Texas.

Let:
- P = You visited Dallas
- Q = You've been to Texas

#### Inverse:
If you did not visit Dallas, then you have not been to Texas.
✘ This is not logically equivalent (false in real life — you could have visited Houston).

#### Converse:
If you've been to Texas, then you visited Dallas.
✘ Not necessarily true — you might have gone to Austin or San Antonio.

#### Contrapositive:
If you have not been to Texas, then you did not visit Dallas.
✔ This is logically equivalent and true.

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2. If m∠P = 90°, then ∠P is a right angle.

Let:
- P = m∠P = 90°
- Q = ∠P is a right angle

This is a definition — so it's always true.

#### Inverse:
If m∠P ≠ 90°, then ∠P is not a right angle.
✔ True (by definition)

#### Converse:
If ∠P is a right angle, then m∠P = 90°.
✔ Also true — this is the reverse definition.

#### Contrapositive:
If ∠P is not a right angle, then m∠P ≠ 90°.
✔ True — logically equivalent.

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3. If it is Saturday, then the school is closed.

Let:
- P = It is Saturday
- Q = School is closed

#### Inverse:
If it is not Saturday, then the school is not closed.
✘ False — schools may be closed on Sundays too, or for holidays.

#### Converse:
If the school is closed, then it is Saturday.
✘ False — schools close on Sundays, holidays, etc.

#### Contrapositive:
If the school is not closed, then it is not Saturday.
✔ True — logically equivalent.

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4. If an angle is a right angle, then it measures 90 degrees.

This is exactly the same as #2, just reversed.

Let:
- P = Angle is a right angle
- Q = It measures 90°

#### Inverse:
If an angle is not a right angle, then it does not measure 90°.
✔ True — by definition

#### Converse:
If an angle measures 90°, then it is a right angle.
✔ True — again, this is the standard definition.

#### Contrapositive:
If an angle does not measure 90°, then it is not a right angle.
✔ True — logically equivalent.

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5. If an animal is a dog, then it has four legs.

Let:
- P = Animal is a dog
- Q = It has four legs

Note: This is not universally true — some dogs may have fewer due to injury. But we assume general case.

#### Inverse:
If an animal is not a dog, then it does not have four legs.
✘ False — cats, horses, etc., have four legs but aren't dogs.

#### Converse:
If an animal has four legs, then it is a dog.
✘ False — many animals have four legs.

#### Contrapositive:
If an animal does not have four legs, then it is not a dog.
✔ True — logically equivalent (if it's not a four-legged animal, it can't be a dog under normal assumptions).

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✔ Final Answers:

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#### 1.
- Inverse: If you did not visit Dallas, then you have not been to Texas.
- Converse: If you've been to Texas, then you visited Dallas.
- Contrapositive: If you have not been to Texas, then you did not visit Dallas.

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#### 2.
- Inverse: If m∠P ≠ 90°, then ∠P is not a right angle.
- Converse: If ∠P is a right angle, then m∠P = 90°.
- Contrapositive: If ∠P is not a right angle, then m∠P ≠ 90°.

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#### 3.
- Inverse: If it is not Saturday, then the school is not closed.
- Converse: If the school is closed, then it is Saturday.
- Contrapositive: If the school is not closed, then it is not Saturday.

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#### 4.
- Inverse: If an angle is not a right angle, then it does not measure 90 degrees.
- Converse: If an angle measures 90 degrees, then it is a right angle.
- Contrapositive: If an angle does not measure 90 degrees, then it is not a right angle.

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#### 5.
- Inverse: If an animal is not a dog, then it does not have four legs.
- Converse: If an animal has four legs, then it is a dog.
- Contrapositive: If an animal does not have four legs, then it is not a dog.

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📌 Summary Tips:

- Always remember: Contrapositive is logically equivalent to the original.
- Converse and Inverse are not guaranteed to be true.
- Use logic carefully — don’t confuse truth with real-world exceptions unless specified.

Let me know if you'd like this formatted into a printable answer sheet!