Is each biconditional statement true or false?
Select True or False for each statement.
True False
A polygon is a square if and only if the polygon has exactly four sides.
A polygon has exactly five sides if and only if the polygon is a pentagon.
A polygon has exactly four sides if and only if the polygon is a rectangle.
To determine the truth value of each biconditional statement, we need to evaluate whether both the "if" part and the "only if" part are true.
1. A polygon is a square if and only if the polygon has exactly four sides.
- To answer this, we need to check if a square has exactly four sides, which is true, and if a polygon with four sides is always a square, which is false. Since both parts are not true, the biconditional statement is false.
2. A polygon has exactly five sides if and only if the polygon is a pentagon.
- We need to verify if a polygon with five sides is always a pentagon, which is true, and if a pentagon always has exactly five sides, which is also true. Since both parts are true, the biconditional statement is true.
3. A polygon has exactly four sides if and only if the polygon is a rectangle.
- We need to check if a polygon with four sides is always a rectangle, which is false, and if a rectangle always has exactly four sides, which is true. Since one part is false, the biconditional statement is false.
Therefore, the answers are:
1. False
2. True
3. False
What if it were a rectangle ?
What if the up and down sides were not all the same length?
What if it had four unequal sides?