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.
