To determine the statement that is logically equivalent to [tex]\( p \rightarrow q \)[/tex], we need to understand the contrapositive of the implication.
Given:
- [tex]\( p \)[/tex]: Angles XYZ and RST are vertical angles.
- [tex]\( q \)[/tex]: Angles XYZ and RST are congruent.
An implication [tex]\( p \rightarrow q \)[/tex] states "If [tex]\( p \)[/tex], then [tex]\( q \)[/tex]". The contrapositive of this statement is logically equivalent to the original implication, and it is formed by:
- Taking the negation of [tex]\( q \)[/tex],
- Taking the negation of [tex]\( p \)[/tex],
- Reversing the direction of the implication.
Therefore, the contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex], which reads "If not [tex]\( q \)[/tex], then not [tex]\( p \)[/tex]".
Let's restate this in terms of our specific p and q:
- [tex]\( \neg q \)[/tex]: Angles XYZ and RST are not congruent.
- [tex]\( \neg p \)[/tex]: Angles XYZ and RST are not vertical angles.
So, the contrapositive statement [tex]\( \neg q \rightarrow \neg p \)[/tex] becomes:
"If angles XYZ and RST are not congruent, then they are not vertical angles."
This statement is logically equivalent to the original statement [tex]\( p \rightarrow q \)[/tex].
Analyzing the given options:
1. If angles XYZ and RST are congruent, then they are vertical angles. (Incorrect)
2. If angles XYZ and RST are not vertical angles, then they are not congruent. (Incorrect)
3. If angles XYZ and RST are not congruent, then they are not vertical angles. (Correct)
4. If angles XYZ and RST are vertical angles, then they are not congruent. (Incorrect)
Therefore, 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."
