Chapter 5.4, Problem 32WE

To determine

To prove: the given theorem.

Explanation of Solution

Given Information:

Given theorem,

Diagonals of rectangle are congruent for rectangle $QRST$ , it is to be shown that $\overline{QS}\cong \overline{RT}$ .

Calculation:

The given statement can be proved as below:

Statement	Reasons
Rectangle $QRST$	Given
$O$ is the midpoint $\overline{RT}$ and $\overline{QS}$	Theorem 5-3 diagonals of a parallelogram bisect each other.
In right $\Delta TQR$, $O$ is the midpoint $\overline{RT}$	Proved above
$\overline{OT}\cong \overline{OQ}\cong \overline{OR}$	Theorem 5-15 the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices
In right $\Delta SRQ$, $O$ is the midpoint $\overline{QS}$	Proved above
$\overline{OS}\cong \overline{OQ}\cong \overline{OR}$	Theorem 5-15 the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices
$\overline{OS}\cong \overline{OT}\cong \overline{OQ}\cong \overline{OR}$	Substitution property
$\begin{array}{l}\overline{OS}\cong \overline{OQ}\cong \overline{OR}\\ \overline{RT}=\overline{OT}+\overline{OR}\end{array}$	Segment addition postulate
$\begin{array}{l}\overline{QS}=2\overline{OQ}\\ \overline{RT}=2\overline{OQ}\end{array}$	Substitution property
$\overline{QS}=\overline{RT}$	Substitution Property

Hence, Theorem 5-12 is proved by substitution, property, segment addition postulate.