To address the question properly, let's break it down step by step:
1. Identify the Original Statement:
- The original statement is: "If a figure is a square, then its diagonals divide it into isosceles triangles."
- Let [tex]\( p \)[/tex] represent "a figure is a square."
- Let [tex]\( q \)[/tex] represent "a figure's diagonals divide it into isosceles triangles."
- The original statement can be written as [tex]\( p \rightarrow q \)[/tex].
2. Determine the Converse of the Statement:
- The converse of the original statement switches the hypothesis and conclusion.
- Therefore, the converse is: "If a figure's diagonals divide it into isosceles triangles, then it is a square."
- Symbolically, this is written as [tex]\( q \rightarrow p \)[/tex].
3. Evaluate the Truth of the Converse:
- To assess the truth value of the converse, we consider geometric properties.
- A square's diagonals indeed divide it into isosceles triangles. However, this property is not unique to squares alone; for instance, rectangles (which are not squares) also have diagonals dividing them into isosceles triangles.
- Thus, the converse statement [tex]\( q \rightarrow p \)[/tex] is not universally true but is sometimes true (when the figure is indeed a square) and sometimes false (when the figure is a non-square rectangle).
4. Select the Correct Answers:
- The statement [tex]\( q \rightarrow p \)[/tex] is the correct representation of the converse of the original statement.
- The truth value of the converse is that it is sometimes true and sometimes false.
Based on this breakdown, the correct answers are:
- [tex]\( q \rightarrow p \)[/tex]
- The converse of the statement is sometimes true and sometimes false.
Incorrect answers and brief justifications:
- [tex]\( p \rightarrow q \)[/tex]: This is the original statement, not the converse.
- The converse of the statement is false: This is incorrect as it is not always false.
- [tex]\( q \leftrightarrow p \)[/tex]: This represents a biconditional statement, which is not the correct converse.
- The converse of the statement is true: This is incorrect as the converse is not universally true.
- [tex]\( \sim p \rightarrow \sim q \)[/tex]: This represents the contrapositive of the original statement, not the converse.
In conclusion, the correct selections are:
- [tex]\( q \rightarrow p \)[/tex]
- The converse of the statement is sometimes true and sometimes false.
