Chapter 14, Problem 6PE

To determine

To prove: a trapezoid cannot have two pairs of congruent sides.

Explanation of Solution

Given information:

  $▭ABCD$ is a trapezoid with diagonal $AC$ , and $BD$ .

  

Calculation:

We will show that $\Delta ABC$ , and $\Delta ABD$ are congruent so that its common sides are the diagonals are also congruent.

So, from $\Delta ABC$ , and $\Delta ABD$ ,

  $AB=AB$ (Common side)

  $\angle DAB=\angle ABC$ (Angles corresponding to side $AB$ )

  $\overline{DB}=\overline{AC}$ (Length of diagonals are equal)

That is by side-angle-side rule it follows that, $\Delta ABC$ and $\Delta ABD$ are congruent. Moreover,

  $\Delta ABC=\Delta ABD$

Hence as a common sides both the diagonals $AC$ and $BD$ are also congruent.

Since $\text{Length of }\overline{DC}\ne \text{Length of }\overline{AB}$ , it is proved that only one pair of sides are congruent.