To solve the problem, we’ll prove the Pythagorean theorem using the provided square and triangles arrangement.

Step 1: Proving $a^2 + b^2 = c^2$

1. Understand the Structure:

   • There’s a large square with side length $a + b$.
   • Inside, there’s a smaller square formed by arranging four identical right triangles around it.

2. Calculate the Area of the Large Square:

   • The side length of the large square is $a + b$.
   • Therefore, its area is: $(a + b)^2 = a^2 + 2ab + b^2$

3. Calculate the Area Using the Inner Square and Triangles:

   • Inside the large square, there’s a smaller square with side length $c$ (the hypotenuse of each right triangle).
   • The area of the inner square is: $c^2$
   • Each of the four triangles has an area of $\frac{1}{2} \times a \times b = \frac{ab}{2}$.
   • The total area of the four triangles is: $4 \times \frac{ab}{2} = 2ab$

4. Equate the Two Area Expressions:

   • The area of the large square can also be written as the sum of the areas of the inner square and the four triangles: $a^2 + 2ab + b^2 = c^2 + 2ab$
   • By subtracting $2ab$ from both sides, we get: $a^2 + b^2 = c^2$
   • This completes the proof of the Pythagorean theorem.

Step 2: Defend the Statement that All Four Triangles Are Congruent

Each of the four triangles has:

• The same leg lengths $a$ and $b$,
• The same hypotenuse length $c$,
• The same right angle.

Since all sides and angles match, the triangles are congruent by the hypotenuse-leg (HL) congruence theorem for right triangles. This confirms that all four triangles are indeed congruent.

Would you like more details or further explanation?

Related Questions for Further Practice:

1. How would you calculate the perimeter of the inner square in terms of $c$?
2. Can you apply the Pythagorean theorem in three dimensions? How would it change?
3. How would the area calculations change if the triangle was an isosceles right triangle?
4. How could this proof be modified if the triangle had sides $a$, $b$, and $d$ such that $a^2 + b^2 \neq d^2$?
5. Can you find another method to prove $a^2 + b^2 = c^2$ using similar triangles?

Tip: Always look for multiple ways to calculate areas or perimeters in a geometric figure; they often reveal relationships between the elements of the shape.