To determine which statement is logically equivalent to [tex]\( p \rightarrow q \)[/tex], we need to consider the contrapositive of the statement. The contrapositive of a conditional statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex], and it is logically equivalent to the original statement. Let’s break it down step-by-step.
Given statements:
- [tex]\( p \)[/tex]: Angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are vertical angles.
- [tex]\( q \)[/tex]: Angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are congruent.
The original statement [tex]\( p \rightarrow q \)[/tex] reads:
"If angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are vertical angles, then angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are congruent."
To find the contrapositive, we negate both [tex]\( q \)[/tex] and [tex]\( p \)[/tex] and switch their order:
- [tex]\( \neg q \)[/tex]: Angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are not congruent.
- [tex]\( \neg p \)[/tex]: Angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are not vertical angles.
The contrapositive [tex]\( \neg q \rightarrow \neg p \)[/tex] reads:
"If angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are not congruent, then angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are not vertical angles."
Let’s compare this to the given answer choices:
1. If angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are congruent, then they are vertical angles.
2. If angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are not vertical angles, then they are not congruent.
3. If angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are not congruent, then they are not vertical angles.
4. If angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are vertical angles, then they are not congruent.
The statement that matches our contrapositive is:
3. If angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are not congruent, then they are not vertical angles.
Therefore, the statement that is logically equivalent to [tex]\( p \rightarrow q \)[/tex] is:
"If angles [tex]\( XYZ \)[/tex] and [tex]\( RST \)[/tex] are not congruent, then they are not vertical angles."
