Chapter 6.5, Problem 37PPS

To determine

The proof of Theorem 6.19.

Explanation of Solution

Given:

The theorem is, “If one pair of consecutive sides of parallelogram is congruent, the parallelogram is a rhombus.”

In a parallelogram ABCD, $\overline{AB}\cong \overline{BC}.$

Calculation:

In a parallelogram, the opposite sides are congruent.

Therefore,

  $\overline{AB}\cong \overline{CD}\text{ and }\overline{\text{BC}}\cong \overline{AD}.$

According to the Transitive Property of Congruence, $\overline{AB}\cong \overline{BC}\cong \overline{\text{CD}}\cong \overline{DA}.$

Hence, all sides of the given parallelogram are congruent.

According to the definition of a rhombus, the given figure is a rhombus.