Question

given: jkl m is a rhombus;
overline{kp} cong overline{jq}; overline{lp} cong overline{kq}
prove: jkl m is a square.

1. overline{jk} cong overline{kl} cong overline{lm} cong overline{jm}
2. overline{kp} cong overline{jq}, overline{lp} cong overline{kq}
3. \triangle kjq cong \triangle lkp

Explanation:

Step1: Properties of a rhombus

In a rhombus, all sides are equal. So, $\overline{JK}\cong\overline{KL}\cong\overline{LM}\cong\overline{JM}$ (given \(JKLM\) is a rhombus).

Step2: Given congruent segments

We are given \(\overline{KP}\cong\overline{JQ}\) and \(\overline{LP}\cong\overline{KQ}\) (from the problem statement).

Step3: Proving triangle congruence

Using the Side - Side - Side (SSS) congruence criterion. In \(\triangle KJQ\) and \(\triangle LKP\), we have:

• \(KJ = LK\) (sides of the rhombus)
• \(JQ=KP\) (given)
• \(KQ = LP\) (given)

So, \(\triangle KJQ\cong\triangle LKP\) (by SSS).

Answer:

1. All sides of a rhombus are equal.
2. Given in the problem.
3. By Side - Side - Side (SSS) congruence criterion.