Answered

Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

Given:
[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.

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

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

Sagot :

GinnyAnswer
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."
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.