Chapter 5.4, Problem 33WE

To determine

To prove: the given theorem for one diagonal of the rhombus.

Explanation of Solution

Given Information:

Given theorem,

For one diagonal of a rhombus bisects two angles of the rhombus.

Prove, $\overline{RT}$ bisects $\angle QRS$ and $\angle QTS$ .

And prove $\overline{QS}$ bisects $\angle TQR$ and $\angle TSR$ .

Calculation:

The given statement can be proved as below:

Statement	Reasons
Rhombus $QRST$, $\overline{QS}$ and $\overline{RT}$ are diagonals	Given
$\overline{QR}\cong \overline{QT}\cong \overline{TS}\cong \overline{SR}$	Definition of rhombus
In $\Delta QRS$ and $\Delta QTS$ $\overline{QR}\cong \overline{QT},\overline{RS}\cong \overline{TS}$ and $\overline{QS}\cong \overline{QS}$	Proved above
$\Delta QRS\cong \Delta QTS$	SSS postulate
$\angle TQS\cong \angle RQS,\angle TSQ\cong \angle TSR$	Definition of Angle Bisector
$\overline{QR}\cong \overline{QT}\cong \overline{TS}\cong \overline{SR}$	Definition of rhombus
In $\Delta TQR$ and $\Delta TSR$ $\overline{TQ}\cong \overline{TS},\overline{QR}\cong \overline{RS}$ and $\overline{TR}\cong \overline{TR}$	Proved above
$\Delta TQR\cong \Delta TSR$	SSS postulate
$\angle QTR\cong \angle STR,\angle QRT\cong \angle SRT$	Parts of congruent triangle are congruent
$\overline{RT}$ bisects $\angle QRS$ and $\angle QTS$.	Definition of angle bisector

Hence, Theorem 5-14 is proved by definition of rhombus, SSS postulate, parts of congruent triangle are congruent, definition of angle bisector, definition of rhombus.