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The image contains a problem where you're asked to prove that triangle \( JKL \) is a right triangle. Here’s a breakdown of the information provided:

1. You have two triangles: \( DEF \) and \( JKL \).
   - \( DF = JL = a \)
   - \( FE = LK = b \)
   - Triangle \( DEF \) is a right triangle, and you are given that the Pythagorean Theorem applies: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.

2. The goal is to prove that triangle \( JKL \) is also a right triangle. Since \( DF \) is equal to \( JL \) and \( FE \) is equal to \( LK \), the side lengths of triangle \( JKL \) should also satisfy the Pythagorean theorem if it is a right triangle.

To complete the proof:
- You need to show that angle \( K \) is a right angle.
- By applying the Pythagorean theorem to triangle \( DEF \) (which is a right triangle) and showing that \( a^2 + b^2 = c^2 \) holds for triangle \( JKL \), you can conclude that triangle \( JKL \) is also a right triangle.

Do you need any further clarifications or detailed steps for the proof?

Here are 5 questions to deepen your understanding:
1. What is the significance of the Pythagorean theorem in determining if a triangle is a right triangle?
2. How do congruent sides between triangles \( DEF \) and \( JKL \) help in proving that both triangles are right triangles?
3. Why is it important to show that the sum of the squares of two sides equals the square of the hypotenuse in triangle \( JKL \)?
4. Could you use any other method besides the Pythagorean theorem to prove that triangle \( JKL \) is a right triangle?
5. What is the relationship between angles and side lengths in a right triangle?

**Tip**: Always check if the sum of the squares of the two shorter sides equals the square of the hypotenuse to determine if a triangle is a right triangle.

In the diagram below, DF = JL = a, FE = LK = b, and triangle DEF is a right triangle. Use the drop-down menus to complete the proof that if the equation a^2 + b^2 = c^2 is true of the side lengths in triangle JKL, then triangle JKL must be a right triangle.

Learn how to prove that triangle JKL is a right triangle by applying the Pythagorean Theorem. This problem involves geometry concepts such as right triangles, congruence, and the Pythagorean Theorem. Suitable for students in Grades 8-10.

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