The worksheet provided focuses on logical reasoning and conditional statements. Here’s how the answers and explanations can be broken down step-by-step:

1. Writing the Conditional Statements

14. Conditional:
A conditional statement takes the form: If p, then q.

• Statement: If two angles are adjacent, then they form a linear pair.
• Truth Value: False (Two adjacent angles do not necessarily form a linear pair; they must also sum to $180^\circ$.)

15. Inverse:
The inverse negates both the hypothesis and the conclusion: If not p, then not q.

• Statement: If two angles are not adjacent, then they do not form a linear pair.
• Truth Value: True

16. Converse:
The converse swaps the hypothesis and conclusion: If q, then p.

• Statement: If two angles form a linear pair, then they are adjacent.
• Truth Value: True

17. Contrapositive:
The contrapositive negates and swaps both: If not q, then not p.

• Statement: If two angles do not form a linear pair, then they are not adjacent.
• Truth Value: True

2. Bi-Conditional Statements

Bi-conditional statements are written as: p if and only if q. They are true only when both the conditional and its converse are true.

18. Lines are perpendicular if and only if they intersect at a right angle.

• Truth Value: True

19. It is midnight if and only if it is dark outside.

• Truth Value: False (It can be dark outside even if it is not midnight.)

20. Triangles are congruent if and only if their corresponding parts are congruent.

• Truth Value: True

3. Laws of Logic

21. Law of Detachment or Syllogism?

• Given: If the kicker makes the field goal, then the team will win the game.
• Given: If the team wins the game, then they will move on to the playoffs.
• Conclusion: If the kicker makes the field goal, then the team will move on to the playoffs.
• Valid By: Law of Syllogism (It connects two conditionals: $p \rightarrow q$ and $q \rightarrow r$, concluding $p \rightarrow r$.)

22. Law of Detachment or Syllogism?

• Given: If a number is a natural number, then it is a whole number.
• Given: $x$ is a natural number.
• Conclusion: $x$ is a whole number.
• Valid By: Law of Detachment (Direct application of $p \rightarrow q$ when $p$ is true.)

Recap and Clarification

Let me know if you need further explanations for any specific part!

Related Questions:

1. How do you determine the truth value of a conditional statement?
2. What is the difference between converse and contrapositive?
3. How can you identify when a statement uses the Law of Syllogism?
4. What are examples where bi-conditional statements are false?
5. What logical errors can occur when interpreting conditional statements?

Tip: Always check the definitions of the hypothesis (p) and conclusion (q) to determine the truth value accurately.