Select all the correct answers.

If a figure is a square, its diagonals divide it into isosceles triangles.

*p*: A figure is a square.

*q*: A figure's diagonals divide into isosceles triangles.

Which represents the converse of this statement? Is the converse true?

 q→p

 The converse of the statement is true.

 q↔p

 p→q

 The converse of the statement is sometimes true and sometimes false.

 ∼p→ ∼q

 The converse of the statement is false.


Let's break down the problem:

1. **Original statement**: "If a figure is a square (*p*), then its diagonals divide it into isosceles triangles (*q*)."
   - This is represented as \( p \rightarrow q \).

2. **Converse of the statement**: The converse of \( p \rightarrow q \) is \( q \rightarrow p \), which translates to: "If a figure's diagonals divide it into isosceles triangles (*q*), then the figure is a square (*p*)."

3. **Truth of the converse**:
   - The converse \( q \rightarrow p \) is **false**. While squares do have diagonals that divide them into isosceles triangles, other figures like rectangles and rhombuses also have this property. Thus, the converse is not always true.

**Correct Answers:**
- \( q \rightarrow p \)
- The converse of the statement is false.

Do you want more details or have any questions?

Here are five related questions to consider:
1. What does it mean for two statements to be logically equivalent?
2. Can you provide an example where \( q \) is true, but \( p \) is false?
3. How would you represent the inverse of the original statement?
4. What is the contrapositive of the original statement?
5. Are there other geometric figures where the diagonals divide them into isosceles triangles?

**Tip:** When evaluating the truth of a converse statement, consider counterexamples where the hypothesis holds, but the conclusion does not.

Explore the converse statement in geometry: If a figure's diagonals divide it into isosceles triangles, does it necessarily mean the figure is a square? Learn why the converse is false and discover related geometric concepts. Ideal for high school students studying geometry.

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