Let's analyze the three statements to determine which are true and which are false.

1. A polygon is a square if and only if the polygon has exactly four sides.

   • False. A polygon having exactly four sides is not sufficient to conclude that the polygon is a square. A square is a specific type of four-sided polygon (quadrilateral) with all sides equal and all angles equal to 90 degrees. The statement is false because not every four-sided polygon (e.g., a rectangle, rhombus, or trapezoid) is a square.

2. A polygon has exactly five sides if and only if the polygon is a pentagon.

   • True. By definition, a polygon with exactly five sides is called a pentagon, and conversely, a pentagon always has exactly five sides. This is a true statement.

3. A polygon has exactly four sides if and only if the polygon is a rectangle.

   • False. A polygon having exactly four sides does not necessarily mean it is a rectangle. A rectangle is a type of quadrilateral where opposite sides are equal, and all angles are 90 degrees. However, other quadrilaterals, such as squares, rhombuses, or trapezoids, also have four sides but are not necessarily rectangles. Therefore, this statement is false.

Summary:

• Statement 1: False
• Statement 2: True
• Statement 3: False

Would you like more details, or do you have any questions? Here are some related questions you might find interesting:

1. What are the necessary conditions for a quadrilateral to be a square?
2. Can a rectangle have unequal sides?
3. What are the different types of quadrilaterals?
4. How does a rhombus differ from a square?
5. What are the properties of a pentagon?

Tip: Always remember that "if and only if" statements require both the condition and its converse to be true.