Practice worksheet on logical reasoning using the Law of Detachment and Law of Syllogism.

Let's solve each problem step by step using logical reasoning, specifically the Law of Detachment and the Law of Syllogism.

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🔹 Recall:

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

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

Now let’s go through each question.

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✔ Part 1: Use Law of Detachment

#### 1.
> If you pass a road test, you will get your driver's license.
> Tamara passed her road test.

- Conditional statement:
If pass road test → get license
- Given fact: Tamara passed the road test → matches the hypothesis.

✔ Conclusion (by Law of Detachment):
Tamara will get her driver's license.

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#### 2.
> If an angle is a right angle, then it is not acute.
> $ m\angle Q \neq 90 $

- Conditional: If right angle → not acute
- Given: $ m\angle Q \neq 90 $ → means ∠Q is not a right angle

But the hypothesis ("if it is a right angle") is not true, so we cannot apply the Law of Detachment.

✘ No conclusion possible.
We only know that ∠Q ≠ 90°, but that doesn’t tell us whether it's acute or not. It could be obtuse or acute — no definite conclusion.

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#### 3.
> If you swim 100 m in less than 90 seconds, you will be on the swim team.
> Jeremy swam 200 m in 190 s.

- Hypothesis: "swim 100 m in <90 sec" → must be true to trigger conclusion.
- But Jeremy swam 200 m in 190 s → his time for 100 m would be about 95 s (190 ÷ 2), which is more than 90 s.

So he did not meet the condition.

✘ No conclusion possible.
We don't know if he qualifies for the team based on this information.

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#### 4.
> If you arrive at the theater by 2:00 p.m., you can see the show.
> Julie arrived at 1:45 p.m.

- Hypothesis: arrive by 2:00 p.m. → she did (1:45 < 2:00)

✔ So by Law of Detachment:
Julie can see the show.

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#### 5.
> If a triangle has one obtuse angle, then it is an obtuse triangle.
> △ABC has an acute angle.

- Conditional: If has one obtuse angle → obtuse triangle
- Given: △ABC has an acute angle → this does not match the hypothesis

We don't know if it has an obtuse angle. A triangle can have acute angles and still be acute, right, or obtuse.

✘ No conclusion possible.
The given fact does not satisfy the hypothesis.

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✔ Part 2: Use Law of Syllogism

#### 6.
> If the sum of the digits in a number is 9, then the number is divisible by 9.
> If a number is divisible by 9, then it is divisible by 3.

- First: If sum of digits = 9 → divisible by 9
- Second: If divisible by 9 → divisible by 3

→ The conclusion of the first (divisible by 9) matches the hypothesis of the second.

✔ Apply Law of Syllogism:
If sum of digits = 9 → divisible by 3

So, final conclusion:
If the sum of the digits in a number is 9, then the number is divisible by 3.

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#### 7.
> If an animal has hair, it is a mammal.
> If an animal gives birth to live young, it is a mammal.

- First: If has hair → mammal
- Second: If gives live birth → mammal

But the conclusion of the first (mammal) is not the hypothesis of the second (live birth).
They both lead to the same conclusion, but are not linked in a chain.

✘ No conclusion possible via Law of Syllogism.
There is no common link between the two statements' hypotheses and conclusions.

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#### 8.
> If a language is Spanish, it is a Romance language.
> If a language is a Romance language, it has Latin roots.

- First: If Spanish → Romance language
- Second: If Romance language → has Latin roots

→ Conclusion of first (Romance) matches hypothesis of second.

✔ Apply Law of Syllogism:
If a language is Spanish, then it has Latin roots.

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#### 9.
> If two angle measures sum to 180, then the angles are supplementary.
> If two angle measures sum to 90, then the angles are complementary.

- First: sum = 180 → supplementary
- Second: sum = 90 → complementary

These are two separate conditions with different sums.

There is no logical connection between them (one is about 180, the other about 90).

✘ No conclusion possible.
The conclusion of one does not match the hypothesis of the other.

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✔ Final Answers:

#### Using Law of Detachment:

1. Yes: Tamara will get her driver's license.
2. No: $ m\angle Q \neq 90 $ does not confirm it's a right angle.
3. No: Jeremy didn't swim 100 m in under 90 s.
4. Yes: Julie can see the show.
5. No: Having an acute angle doesn't imply it has an obtuse angle.

#### Using Law of Syllogism:

6. Yes: If sum of digits is 9, then the number is divisible by 3.
7. No: No common link between the statements.
8. Yes: If a language is Spanish, then it has Latin roots.
9. No: The statements are unrelated (different sums).

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Let me know if you'd like these written in a table format!