Question

2 given: \\(\overline{jl} \cong \overline{nm}\\), \\(k\\) is the midpoint of \\(\overline{jn}\\) and \\(\overline{lm}\\) prove: \\(\triangle jkl \cong \triangle nkm\\) \\(\

$$\begin{array}{|c|c|} \\hline \\text{statements} & \\text{reasons} \\\\ \\hline 1. \\ \\overline{jl} \\cong \\overline{nm} & 1. \\\\ \\hline 2. \\ k \\text{ is the midpoint of } \\overline{jn} \\text{ and } \\overline{lm} & 2. \\\\ \\hline 3. \\ \\overline{jk} \\cong \\overline{nk} & 3. \\\\ \\hline 4. \\ \\overline{lk} \\cong \\overline{mk} & 4. \\\\ \\hline 5. \\ \\triangle jkl \\cong \\triangle nkm & 5. \\\\ \\hline \\end{array}$$

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Explanation:

Step1: State given congruent segment

$\overline{JL} \cong \overline{NM}$

Step2: State given midpoint info

$K$ is the midpoint of $\overline{JN}$ and $\overline{LM}$

Step3: Use midpoint definition for $\overline{JN}$

A midpoint divides a segment into two congruent parts, so $\overline{JK} \cong \overline{NK}$

Step4: Use midpoint definition for $\overline{LM}$

A midpoint divides a segment into two congruent parts, so $\overline{LK} \cong \overline{MK}$

Step5: Apply SSS congruence rule

We have three pairs of congruent sides: $\overline{JL} \cong \overline{NM}$, $\overline{JK} \cong \overline{NK}$, $\overline{LK} \cong \overline{MK}$, so $\triangle JKL \cong \triangle NKM$

Answer:

Statements	Reasons
2. $K$ is the midpoint of $\overline{JN}$ and $\overline{LM}$	2. Given
3. $\overline{JK} \cong \overline{NK}$	3. Definition of midpoint
4. $\overline{LK} \cong \overline{MK}$	4. Definition of midpoint
5. $\triangle JKL \cong \triangle NKM$	5. Side-Side-Side (SSS) Congruence Postulate