Chapter 6.3, Problem 18WE

To determine

To prove: that the diagonals of a trapezoid do not bisect each other.

Explanation of Solution

Consider a trapezoid $ABCD$ . The diagonals of trapezoid intersect each other at $O$ .

An indirect proof is initiated by assuming temporarily that whatever is need to prove is untrue and then work from there to finally conclude that the assumption is untrue.

Proof:

Assume temporarily that the diagonals of the trapezoid bisect each other, that is

  $AO=OC$ and $DO=OB$ .

Therefore, $\Delta s$ , $AOB$ and $DOC$ are congruent by the SAS axiom.

Then the corresponding parts $AB$ and $DC$ of these congruent triangles will be congruent.

However, this would mean that the non −parallel sides of the trapezoid are equal, but by definition, the non-parallel sides of a trapezoid are not equal.

Hence, the initial assumption that the diagonals of a trapezoid bisect each other is not true.

Therefore, the diagonals of a trapezoid do not bisect each other.