Question

Prove Theorem 5-12.

Short answer

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

Step-by-step solution

Step 1. State the theorem 5-12.

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

Step 2. State the proof.

The given statement can be proved as below:

Statement	Reasons
Rectangle $QRST$.	Given.
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.

Step 3. State the conclusion.

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