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?

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!