Complete Two-Column Proof: Prove $ riangle JMK \cong riangle LKM$ with Given Congruent Sides and Angles

Question

Given: $\overline{JK} \cong \overline{LM}$, $\angle JKM \cong \angle LMK$; Prove: $	riangle JMK \cong 	riangle LKM$. Complete the two-column proof with statements and reasons.

Accepted Answer

Completed two-column proof as follows: 
Statements: 
1. $\overline{JK} \cong \overline{LM}$ 
2. $\angle JKM \cong \angle LMK$ 
3. $\overline{KM} \cong \overline{MK}$ 
4. $	riangle JMK \cong 	riangle LKM$

Reasons: 
1. Given 
2. Given 
3. Reflexive Property of Congruence 
4. Side-Angle-Side (SAS) Congruence Postulate

Answer

Step 1: List the given information as the first two statements and reasons. The problem states $\overline{JK} \cong \overline{LM}$ and $\angle JKM \cong \angle LMK$, so these are the first two entries with 'Given' as their reasons. Step 2: Identify the shared side between the two triangles. $\overline{KM}$ is a side of both $	riangle JMK$ and $	riangle LKM$. By the Reflexive Property of Congruence, any segment is congruent to itself, so $\overline{KM} \cong \overline{MK}$ is the third statement with this property as the reason. Step 3: Apply the SAS Congruence Postulate. We have a pair of congruent sides ($\overline{JK} \cong \overline{LM}$), a pair of congruent included angles ($\angle JKM \cong \angle LMK$), and another pair of congruent sides ($\overline{KM} \cong \overline{MK}$). This satisfies the conditions for SAS, so we conclude $	riangle JMK \cong 	riangle LKM$ as the fourth statement with the SAS postulate as the reason.