To solve this problem, we need to analyze the given statements and their logical relationships.
We start with the given propositions:
- [tex]\( p \)[/tex]: "A figure is a square."
- [tex]\( q \)[/tex]: "A figure's diagonals divide into isosceles triangles."
The original implication provided is [tex]\( p \rightarrow q \)[/tex], which reads as: "If a figure is a square, then its diagonals divide it into isosceles triangles." This is given and understood to be true.
Step 1: Identify the converse of the implication
The converse of an implication [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex].
So, the converse of "If a figure is a square, then its diagonals divide it into isosceles triangles" is "If a figure's diagonals divide into isosceles triangles, then the figure is a square."
Step 2: Determine the truth of the converse statement
We need to check if the converse [tex]\( q \rightarrow p \)[/tex] is always true. Let's analyze:
A square's diagonals do indeed divide it into isosceles triangles, but this property is not exclusive to squares. For example, rhombuses also have diagonals that divide the figure into isosceles triangles. This means that while the converse may sometimes hold true (for squares), it does not always hold true because other shapes can also satisfy [tex]\( q \)[/tex] without being squares.
Therefore, the converse [tex]\( q \rightarrow p \)[/tex] is not always true. Thus, the correct conclusion is that the converse is false in general.
Step 3: Match the results to the options provided
- The converse of the statement is sometimes true and sometimes false.
- The converse of the statement is true.
- [tex]\( q \leftrightarrow p \)[/tex]
- [tex]\( q \rightarrow p \)[/tex]
- The converse of the statement is false.
- [tex]\( \sim p \rightarrow \sim q \)[/tex]
- [tex]\( p \rightarrow q \)[/tex]
Correct Answers:
- The converse of the statement represented as [tex]\( q \rightarrow p \)[/tex].
- The converse of the statement is false.
From the analysis, it is clear that the correct answers are:
- [tex]\( q \rightarrow p \)[/tex] (Option 3) which describes the converse of the original statement.
- The converse of the statement is false (Option 5).
Therefore, the correct selections are:
- [tex]\( q \rightarrow p \)[/tex] (Option 4 in the provided listing)
- The converse of the statement is false (Option 5 in the provided listing)
