Question

read the false statement below and then tell whether the converse, inverse, and contrapositive are true. if a figure is a polygon, then it is a quadrilateral. converse inverse contrapositive if a figure is a quadrilateral, then it is a polygon.

Explanation:

Step1: Analyze the converse

A quadrilateral is a type of polygon. So, if a figure is a quadrilateral, then it is a polygon. The converse is true.

Step2: Analyze the inverse

The original statement is “If a figure is a polygon, then it is a quadrilateral”. The inverse is “If a figure is not a polygon, then it is not a quadrilateral”. A non - polygon cannot be a quadrilateral (since a quadrilateral is a polygon). But the original statement is false (because there are polygons that are not quadrilaterals like triangles). However, the inverse is “If not \(p\) (not a polygon), then not \(q\) (not a quadrilateral)”. Since \(q\) (being a quadrilateral) implies \(p\) (being a polygon) (from the converse which is true), by the law of contrapositive for the converse (the inverse of the original is the contrapositive of the converse), the inverse is false.

Step3: Analyze the contrapositive

The contrapositive of the original statement “If \(p\) (a figure is a polygon) then \(q\) (it is a quadrilateral)” is “If not \(q\) (a figure is not a quadrilateral), then not \(p\) (it is not a polygon)”. But there are non - quadrilateral polygons (e.g., triangles). So the contrapositive is false.

Answer:

Converse: True; Inverse: False; Contrapositive: False