Use representations to communicate mathematical ideas. Given JK ≅ LM, JM ≅ LK, prove ΔJKM ≅ ΔLMK. Fill in the proof chart for parts a and b.

Understand the Problem

The question is asking to complete a proof using geometric relationships and given information about the sides of triangles. It requires identifying appropriate lengths or segments to show the congruence of triangles JKM and LKM.

Answer

The completed proof shows that triangles $JKM$ and $LKM$ are congruent: - a. $JM \cong LK$ - b. $KM \cong KM$ Thus, $$ \triangle JKM \cong \triangle LKM $$

Steps to Solve

Identify Given Information We have the following given lengths:

$JK \cong LM$
$JM \cong LK$

Establish Required Lengths for Congruence To prove that triangles $JKM$ and $LKM$ are congruent, we need to establish one more pair of congruent sides. We will use the fact that $KM \cong KM$ (reflexive property).
Fill in the Flow Chart Using the information provided:

For part a, we have $JM \cong LK$ as one congruence.
For part b, we state $KM \cong KM$ from reflexive property.

Apply Side-Side-Side (SSS) Postulate Since we have:

$JK \cong LM$
$JM \cong LK$
$KM \cong KM$

We can conclude by the SSS postulate that: $$ \triangle JKM \cong \triangle LKM $$

Complete the Proof Therefore, complete the proof that triangles $JKM$ and $LKM$ are congruent:

a. $JM \cong LK$
b. $KM \cong KM$

The completed proof shows that triangles $JKM$ and $LKM$ are congruent:

a. $JM \cong LK$
b. $KM \cong KM$

Thus, $$ \triangle JKM \cong \triangle LKM $$

More Information

The SSS (Side-Side-Side) postulate states that if three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. This property is fundamental in proving congruence in geometry.

Tips

Misidentifying the sides that are equal can lead to incorrect conclusions about triangle congruence. Always ensure to use the given information accurately.
Forgetting to note the reflexive property for sides that are identical in both triangles.

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