Answered

Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

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

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

A. If angles XYZ and RST are congruent, then they are vertical angles.
B. If angles XYZ and RST are not vertical angles, then they are not congruent.
C. If angles XYZ and RST are not congruent, then they are not vertical angles.
D. If angles XYZ and RST are vertical angles, then they are not congruent.

Sagot :

GinnyAnswer
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."
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.