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The task is to prove the Pythagorean theorem, which states that for a right triangle with leg lengths \(a\) and \(b\), and hypotenuse length \(c\), the equation \(a^2 + b^2 = c^2\) holds.

Let's break down the proof step-by-step:

1. **Identify the Triangle's Properties:**
   - Since we have a right triangle, one of the angles is \(90^\circ\).
   - The triangle has two legs (labeled as \(a\) and \(b\)) and a hypotenuse (labeled as \(c\)).

2. **Angle Sum Property of a Triangle:**
   - The sum of the interior angles in any triangle is \(180^\circ\).
   - For this triangle, the angles are labeled as \(x\), \(y\), and \(90^\circ\).
   - Therefore, we can state:
     \[
     x + y + 90 = 180
     \]
   - Simplifying this, we get:
     \[
     x + y = 90
     \]
   - This confirms that the two non-right angles \(x\) and \(y\) are complementary (they add up to \(90^\circ\)).

3. **Using the Pythagorean Theorem:**
   - The Pythagorean theorem applies directly to right triangles and is defined as:
     \[
     a^2 + b^2 = c^2
     \]
   - To prove this theorem, you could use geometric methods, such as creating two squares, one with side \(a + b\) (containing four right triangles and a square of side \(c\) inside it), and rearranging areas. Alternatively, trigonometric or algebraic approaches can also validate this relationship.

4. **Conclusion:**
   - This theorem holds universally for all right triangles and is foundational in Euclidean geometry.

For your response, you can write:

**Answer:**
To defend the statement \(x + y + 90 = 180\):
The sum of angles in a triangle is always \(180^\circ\). Since this is a right triangle with one angle equal to \(90^\circ\), the other two angles, \(x\) and \(y\), must add up to \(90^\circ\) to satisfy the angle sum property. Therefore, \(x + y + 90 = 180\) is correct.

Would you like further details on proving the Pythagorean theorem using geometric or algebraic methods?

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**Related Questions:**

1. What are some geometric proofs of the Pythagorean theorem?
2. How does the Pythagorean theorem apply to coordinate geometry?
3. How can we use the Pythagorean theorem in real-life scenarios?
4. What are trigonometric ratios, and how do they relate to right triangles?
5. Can we use the Pythagorean theorem in non-Euclidean geometry?

**Tip:** When solving problems involving right triangles, always check if the Pythagorean theorem applies—it’s a quick way to verify side lengths!

Given: A right triangle has leg lengths a and b and hypotenuse length c. Prove: a^2 + b^2 = c^2

Explore a step-by-step proof of the Pythagorean theorem for a right triangle with legs a and b, and hypotenuse c. Understand the relationship between the triangle's angles and sides, covering the core principles of geometry and algebra. Suitable for Grades 8-10.

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