Chapter 6.5, Problem 35PPS

To determine

To prove: The given theorem.

Explanation of Solution

Given information:

The theorem: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

The diagram:

  

Formula used:

The opposite angles of a parallelogram are congruent.

The sides of a rhombus are congruent.

SAS congruence:

If two pairs of corresponding sides and one pair of corresponding angles of two triangles are congruent then the two angles are congruent.

In the triangles JKL and JML;

  $JM\cong KL$ and $JK\cong LM$ [Opposite sides of a parallelogram are congruent]

  $\angle JKL\cong \angle JML$ [Opposite angles of a parallelogram are congruent]

Thus, the triangles JKL and JML are congruent.

Similarly, the triangles KLM and KJM are also congruent.

Since, the triangles are congruent;

So, $JK\cong KL\cong LM\cong MJ$

It is known that all sides of a rhombus are congruent.

Thus, the parallelogram JKLM is a rhombus.

Hence, the theorem is proved.