Chapter 5.7, Problem 27PSC

To determine

To prove: the two of the triangles formed by diagonals in a quadrilateral are isosceles.

Explanation of Solution

Given information:

The diagonals of a quadrilateral are congruent.

Exactly one pair of opposite sides are congruent.

Formula used:

Properties of parallelogram.

Proof:

  

Let the quadrilateral be shown in the above figure.

Given that $\overline{AB}\cong \overline{DC}\text{ and }{d}_1\cong d_2$ implies that $ABCD$ is a parallelogram.

In parallelogram $ABCD$ , diagonals bisect each other at $O$ .

  $d_1\cong d_2,AO=OC=BO=OD=\frac{d}{2}$

In $ABCD$ :

  $\Delta ABO$ is isosceles triangle since $AO=BO$

  $\Delta CDO$ is isosceles triangle since $DO=CO$