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The problem involves using the isosceles triangle theorem, which states: *If two sides of a triangle are congruent, then the angles opposite those sides are congruent.* 

Given the diagram and the multiple-choice options, we need to decide on the best strategy to prove that the two triangles \( \triangle JKM \) and \( \triangle LKM \) are congruent.

### Key Steps in Understanding the Question:
- The triangle \( \triangle JKL \) is isosceles because \( JK \cong LK \) (marked as congruent).
- We want to use the appropriate method to prove the congruence of \( \triangle JKM \) and \( \triangle LKM \), either by using **SSS (side-side-side)**, **SAS (side-angle-side)**, or **ASA (angle-side-angle)**.

### Analyze Each Option:

- **Option A:** This suggests drawing \( KM \) such that \( M \) is the midpoint of \( JL \) and then proving \( \triangle JKM \cong \triangle LKM \) using the **SSS** congruence theorem. However, \( KM \) is not necessarily equal to \( KM \) for both triangles, so SSS might not be the best strategy here.

- **Option B:** This suggests drawing \( KM \) with \( M \) as the midpoint and using **SAS**. Since the sides \( JK \cong LK \) and we can define an angle between them, SAS might be a valid approach.

- **Option C:** This involves bisecting \( \angle K \) and using **ASA**. This is a possible method since we know two angles and the included side, which fits the ASA criteria.

- **Option D:** This is another variation of ASA but not a bisected angle.

### Correct Answer:
- **Option C**: Drawing \( KM \) bisecting \( \angle K \), with \( M \) on \( JL \), then proving \( \triangle JKM \cong \triangle LKM \) using **ASA** is the best strategy. This is because we can use the angle bisector to guarantee two equal angles and the shared side \( KM \), which satisfies the ASA congruence rule.

Would you like further details on why ASA works best, or do you have any other questions?

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### Related Questions:
1. How does the Angle-Side-Angle (ASA) theorem prove triangle congruence?
2. What are the common theorems used for proving congruence in triangles?
3. Why is the midpoint of a side important in isosceles triangles?
4. What role does the angle bisector play in proving congruence?
5. How does the isosceles triangle theorem help in geometric proofs?

### Tip:
When proving triangles congruent, always look for shared sides, angle bisectors, and symmetry—these can often help in applying the right theorem (SSS, SAS, or ASA).

If you are using this figure to prove the isosceles triangle theorem, which of the following would be the best strategy?

Explore how to prove the isosceles triangle theorem using triangle congruence strategies such as ASA, SSS, and SAS. The problem focuses on determining the best method to prove the congruence of triangles in an isosceles triangle.

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