Topic
Given the following proposition of the form \(P \Rightarrow Q\): “If quadrilateral ABCD is a parallelogram, then quadrilateral ABCD has two diagonals that intersect at the midpoint of each”
a) Is the above statement true or false?
b) State the inverse of the above proposition and consider the true and false of that inverse.
Solution method – See details
a) The following proposition \(P \Rightarrow Q\) is false only when P is true and Q is false; true in the remaining cases
b) The inverse of \(P \Rightarrow Q\) is \(Q \Rightarrow P\).
Detailed explanation
a) The proposition \(P \Rightarrow Q\) is true (based on properties of parallelograms)
b) Inverse proposition: “If quadrilateral ABCD has two diagonals intersecting at the midpoint of each line, then quadrilateral ABCD is a parallelogram”
Is a true statement (based on the sign of parallelogram recognition).
