Consider the statement, “All quadrilaterals have four sides.” Its inverse is _.
The Correct Answer and Explanation is:
Correct Answer:
The inverse of the statement “All quadrilaterals have four sides” is:
“All figures that do not have four sides are not quadrilaterals.”
Explanation:
To form the inverse of a conditional statement, you negate both the hypothesis and the conclusion of the original statement.
The original statement is:
“All quadrilaterals have four sides.”
This can be rewritten in the standard “if-then” format to make it easier to analyze:
“If a figure is a quadrilateral, then it has four sides.”
Here,
- Hypothesis (P): “A figure is a quadrilateral”
- Conclusion (Q): “The figure has four sides”
So the inverse is:
“If a figure is not a quadrilateral, then it does not have four sides.”
But to match the structure of the original “all” statement, we rewrite it:
“All figures that do not have four sides are not quadrilaterals.”
This keeps the logical structure consistent while expressing the negation of both parts.
Let us now test the reasoning:
- Original: Every quadrilateral must have four sides.
- Inverse: If something lacks four sides, it cannot be a quadrilateral.
This makes logical sense, because having four sides is essential to being a quadrilateral. Without four sides, a shape cannot meet the definition of a quadrilateral.
It’s important to understand that the inverse is not always logically equivalent to the original statement in general logic. However, in this specific case, the inverse happens to be true, because the definition of a quadrilateral is a polygon with four sides. So, a shape without four sides cannot be a quadrilateral.
Understanding how to construct the inverse is essential in both geometry and logic, as it helps in forming valid arguments and exploring different truth relationships between statements.
