Question

19.) given: \\( \overline{jk} \cong \overline{lk} \\), \\( \overline{jm} \cong \overline{lm} \\) prove: \\( \triangle kjm \cong \triangle klm \\) \
(table with statements and reasons columns, rows 1 - 4; image of triangle with points k, j, l, m)

Explanation:

Step1: Identify Given Information

We know \( \overline{JK} \cong \overline{LK} \) and \( \overline{JM} \cong \overline{LM} \), and \( \overline{KM} \) is common to both \( \triangle KJM \) and \( \triangle KLM \). So, \( \overline{KM} \cong \overline{KM} \) (Reflexive Property of Congruence).

Step2: Apply SSS Congruence Criterion

In \( \triangle KJM \) and \( \triangle KLM \):

• \( \overline{JK} \cong \overline{LK} \) (Given)
• \( \overline{JM} \cong \overline{LM} \) (Given)
• \( \overline{KM} \cong \overline{KM} \) (Reflexive Property)

By the Side - Side - Side (SSS) Congruence Postulate, if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. So, \( \triangle KJM \cong \triangle KLM \) by SSS.

Now, filling the table:

Statements Column:

1. \( \overline{JK} \cong \overline{LK} \), \( \overline{JM} \cong \overline{LM} \) (Given)
2. \( \overline{KM} \cong \overline{KM} \) (Reflexive Property of Congruence)
3. \( \triangle KJM \) and \( \triangle KLM \) have \( \overline{JK} \cong \overline{LK} \), \( \overline{JM} \cong \overline{LM} \), \( \overline{KM} \cong \overline{KM} \)
4. \( \triangle KJM \cong \triangle KLM \)

Reasons Column:

1. Given
2. Reflexive Property of Congruence
3. From Steps 1 and 2
4. SSS Congruence Postulate

Answer:

To prove \( \triangle KJM\cong\triangle KLM \):

Statements:

1. \( \overline{JK} \cong \overline{LK} \), \( \overline{JM} \cong \overline{LM} \)
2. \( \overline{KM} \cong \overline{KM} \)
3. Three sides of \( \triangle KJM \) are congruent to three sides of \( \triangle KLM \)
4. \( \triangle KJM \cong \triangle KLM \)

Reasons:

1. Given
2. Reflexive Property of Congruence
3. From Statements 1 and 2
4. SSS (Side - Side - Side) Congruence Postulate

(The final answer for the congruence proof is that \( \triangle KJM\cong\triangle KLM \) by the SSS Congruence Postulate.)