Geometry Practice On Law of Detachment and Law of Syllogism | PDF ... - Free Printable

Here's the complete solution to all 10 problems, applying Law of Detachment and Law of Syllogism as appropriate.

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🔹 Key Definitions:

- Law of Detachment:
If you have:
> If P, then Q.
> P is true.
→ Then you can conclude: Q is true.

- Law of Syllogism:
If you have:
> If P, then Q.
> If Q, then R.
→ Then you can conclude: If P, then R.

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## ✔ Problem 1:
> If I go to the movie, then I’ll eat popcorn.
> If I eat popcorn, then I’ll enjoy the movie.

A. Conclusion?
→ *If I go to the movie, then I’ll enjoy the movie.*

B. Law of logic?
→ Law of Syllogism
*(P → Q, Q → R ⇒ P → R)*

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## ✔ Problem 2:
> If a figure is a square, then it is a rectangle.
> If a figure is a rectangle, then it has 4 right angles.

A. Conclusion?
→ *If a figure is a square, then it has 4 right angles.*

B. Law of logic?
→ Law of Syllogism

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## ✔ Problem 3:
> If I miss my bus, then I’ll be late for school.
> I miss my bus.

A. Conclusion?
→ *I’ll be late for school.*

B. Law of logic?
→ Law of Detachment
*(P → Q, P ⇒ Q)*

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## ✔ Problem 4:
> If this wind keeps up, then we will lose some trees.
> We lose some trees.

A. Conclusion?
→ *No valid conclusion can be made.*

B. Law of logic?
→ *None — This is the Fallacy of the Converse.*
You cannot conclude the *if* part (wind keeps up) just because the *then* part happened.
*(P → Q, Q ⇒ no conclusion about P)*

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## ✔ Problem 5:
> All accountants enjoy mathematics.
> People who enjoy mathematics are intelligent.

A. Conclusion?
→ *All accountants are intelligent.*

B. Law of logic?
→ Law of Syllogism
*(Accountant → enjoys math; enjoys math → intelligent ⇒ Accountant → intelligent)*

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## ✔ Problem 6:
> If a person is a librarian, then they read books.
> If a person is a friend of Dana’s, then they read books.

A. Conclusion?
→ *No valid conclusion can be made.*

B. Law of logic?
→ *None — This is not syllogism or detachment.*
Both conditionals share the same conclusion (“read books”), but there’s no chain from librarian → friend of Dana or vice versa. So no logical inference between the two groups.

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## ✔ Problem 7:
> If an animal is a pig, then it is a mammal.
> Leon is a pig.

A. Conclusion?
→ *Leon is a mammal.*

B. Law of logic?
→ Law of Detachment

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## ✔ Problem 8:
> If a quadrilateral is a square, then it has four congruent sides.
> Quadrilateral ABCD has four congruent sides.

A. Conclusion?
→ *No valid conclusion can be made.*

B. Law of logic?
→ *None — Fallacy of the Converse.*
Having 4 congruent sides doesn’t mean it’s a square — it could be a rhombus.
*(P → Q, Q ⇒ no conclusion about P)*

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## ✔ Problem 9:
> If my mother lets me go to a movie, then I will be happy.
> My mom lets me go to a movie.

A. Conclusion?
→ *I will be happy.*

B. Law of logic?
→ Law of Detachment

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## ✔ Problem 10:
> If a man is wealthy, then he buys a mansion.
> If a man can buy a mansion, then he can hire a housekeeper.

A. Conclusion?
→ *If a man is wealthy, then he can hire a housekeeper.*

B. Law of logic?
→ Law of Syllogism
*(Wealthy → buys mansion; buys mansion → hires housekeeper ⇒ Wealthy → hires housekeeper)*

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## 📝 Final Answer Summary:

| # | Conclusion | Law of Logic |
|---|--------------------------------------------------|----------------------|
| 1 | If I go to the movie, then I’ll enjoy the movie. | Law of Syllogism |
| 2 | If a figure is a square, then it has 4 right angles. | Law of Syllogism |
| 3 | I’ll be late for school. | Law of Detachment |
| 4 | No valid conclusion | None (Fallacy) |
| 5 | All accountants are intelligent. | Law of Syllogism |
| 6 | No valid conclusion | None |
| 7 | Leon is a mammal. | Law of Detachment |
| 8 | No valid conclusion | None (Fallacy) |
| 9 | I will be happy. | Law of Detachment |
|10 | If a man is wealthy, then he can hire a housekeeper. | Law of Syllogism |

✔ You’re all set! Let me know if you’d like visual diagrams or more practice!