Which statement can be written as a biconditional statement?
A. If a polygon has 4 sides, then the figure is a quadrilateral.
B. If an angle measures $46^{\circ}$, then it is an acute angle.
C. If $x=-4$, then $x^2=16$.
D. If two angles are supplementary, then one is obtuse and one is acute.
Answer (1)
A biconditional statement is true if both the conditional statement and its converse are true.
Analyze each statement by writing its conditional and converse.
Determine if both the conditional and converse are true for each statement.
Only the statement 'If a polygon has 4 sides, then the figure is a quadrilateral' can be written as a biconditional statement because both the conditional and converse are true.
Explanation
Analyze the statements Let's analyze each statement to determine if it can be written as a biconditional statement. A biconditional statement is true if both the original conditional statement and its converse are true.
Statement 1 analysis Statement 1: If a polygon has 4 sides, then the figure is a quadrilateral.
Conditional: If a polygon has 4 sides, then the figure is a quadrilateral. This is true.
Converse: If a figure is a quadrilateral, then the polygon has 4 sides. This is also true. Since both the conditional and converse are true, this statement can be written as a biconditional.
Statement 2 analysis Statement 2: If an angle measures 4 6 ∘ , then it is an acute angle.
Conditional: If an angle measures 4 6 ∘ , then it is an acute angle. This is true.
Converse: If an angle is an acute angle, then it measures 4 6 ∘ . This is false, as an acute angle can have any measure between 0 ∘ and 9 0 ∘ .
Since the converse is false, this statement cannot be written as a biconditional.
Statement 3 analysis Statement 3: If x = − 4 , then x 2 = 16 .
Conditional: If x = − 4 , then x 2 = 16 . This is true since ( − 4 ) 2 = 16 .
Converse: If x 2 = 16 , then x = − 4 . This is false, as x could also be 4 since 4 2 = 16 .
Since the converse is false, this statement cannot be written as a biconditional.
Statement 4 analysis Statement 4: If two angles are supplementary, then one is obtuse and one is acute.
Conditional: If two angles are supplementary, then one is obtuse and one is acute. This is false. For example, two right angles ( 9 0 ∘ each) are supplementary, but neither is obtuse nor acute. Since the conditional is false, this statement cannot be written as a biconditional.
Conclusion Therefore, only the first statement can be written as a biconditional statement.
Examples
Biconditional statements are useful in defining mathematical concepts precisely. For example, in geometry, we can say that a polygon is a quadrilateral if and only if it has four sides. This means that if a polygon is a quadrilateral, it must have four sides, and if a polygon has four sides, it must be a quadrilateral. This type of precise definition is essential for building a consistent and logical mathematical system.
