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Given:
[tex]\[ p: \text{Angles } XYZ \text{ and } RST \text{ are vertical angles.} \][/tex]
[tex]\[ q: \text{Angles } XYZ \text{ and } RST \text{ are congruent.} \][/tex]

Which statement is logically equivalent to [tex]\( p \rightarrow q \)[/tex]?

A. If angles [tex]\( XYZ \)[/tex] and RST are congruent, then they are vertical angles.
B. If angles [tex]\( XYZ \)[/tex] and RST are not vertical angles, then they are not congruent.
C. If angles [tex]\( XYZ \)[/tex] and RST are not congruent, then they are not vertical angles.
D. If angles [tex]\( XYZ \)[/tex] and RST are vertical angles, then they are not congruent.



Answer :

GinnyAnswer
To determine which statement is logically equivalent to [tex]\( p \rightarrow q \)[/tex] (if [tex]\( p \)[/tex] then [tex]\( q \)[/tex]), we need to identify the contrapositive of the given statement. The contrapositive of a conditional statement [tex]\( p \rightarrow q \)[/tex] is logically equivalent to the original statement [tex]\( p \rightarrow q \)[/tex].

1. Given statements:
[tex]\( p \)[/tex]: Angles XYZ and RST are vertical angles.
[tex]\( q \)[/tex]: Angles XYZ and RST are congruent.

2. Original statement:
[tex]\( p \rightarrow q \)[/tex]: If angles XYZ and RST are vertical angles, then they are congruent.

3. Contrapositive:
The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex].
This means: If angles XYZ and RST are not congruent, then they are not vertical angles.

4. Analyzing each of the provided statements:
- Statement 1: If angles XYZ and RST are congruent, then they are vertical angles.
- This is the converse of the original statement (not equivalent).

- Statement 2: If angles XYZ and RST are not vertical angles, then they are not congruent.
- This is the inverse of the original statement (not equivalent).

- Statement 3: If angles XYZ and RST are not congruent, then they are not vertical angles.
- This is the contrapositive of the original statement (logically equivalent).

- Statement 4: If angles XYZ and RST are vertical angles, then they are not congruent.
- This statement is the negation of the original statement (not equivalent).

5. Conclusion:
The statement that is logically equivalent to [tex]\( p \rightarrow q \)[/tex] is: If angles XYZ and RST are not congruent, then they are not vertical angles.

Thus, the correct choice is statement 3.

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