Question

Prove Theorem 5-3.

Short answer

It is proved that the diagonals of a parallelogram bisect each other.

Step-by-step solution

Step 1. Draw and label the diagram.

Step 2. Apply property of parallelogram.

The opposite sides of a parallelogram are congruent.

In $▱QRST$, $\overline{QR}$and $\overline{TS}$are opposite sides. Therefore, $\overline{QR}\cong \overline{TS}$.

Step 3. Description of step.

From the figure, it can be observed that $\overline{QR}\parallel \overline{TS}$and $\overline{TR}$is a transversal then $\angle STM$and $\angle MRQ$are alternate interior angles such that, $\angle STM\cong \angle MRQ$.

From the figure, it can be observed that $\overline{QR}\parallel \overline{TS}$and $\overline{SQ}$is a transversal then $\angle TSM$and $\angle MQR$are alternate interior angles such that, $\angle TSM\cong \angle MQR$.

Step 4. Description of step.

As $\angle STM\cong \angle MRQ$, $\overline{QR}\cong \overline{TS}$and $\angle TSM\cong \angle MQR$then by ASA postulate $\Delta QMR\cong \Delta SMT$.

Step 5. Description of step.

As $\angle TSM\cong \angle MQR$then by corresponding parts of congruent triangles, $\overline{QM}\cong \overline{MS}$and $\overline{RM}\cong \overline{MT}$.

It is given that $\overline{QS}$and $\overline{TR}$are the diagonals of a parallelogram. From $\overline{QM}\cong \overline{MS}$and $\overline{RM}\cong \overline{MT}$ it can be concluded that the diagonals of a parallelogram bisect each other at point M.

Hence it is proved that diagonals of a parallelogram bisect each other.