Let's solve the problem step by step based on the content provided in the image.

Part 3 — Algebra to Complete the Proof:

1. Step 1:
   Multiply both sides of Proportion 1 by $ac$.

   • Proportion 1 is likely $\frac{e}{a} = \frac{f}{c}$.
   • Multiplying both sides by $ac$:
     $e \cdot c = f \cdot a$

2. Step 2:
   Multiply both sides of Proportion 2 by $bc$.

   • Proportion 2 is likely $\frac{d}{b} = \frac{f}{c}$.
   • Multiplying both sides by $bc$:
     $d \cdot c = f \cdot b$

3. Step 3:
   Add the two resulting equations:
   $e \cdot c + d \cdot c = f \cdot a + f \cdot b$
   Factoring the right-hand side:
   $c(e + d) = f(a + b)$

4. Step 4:
   Use the Segment Addition Postulate, which states that $e + d = c$.
   Therefore, the equation becomes:
   $c^2 = f^2$

Problem 11 — Error Analysis (Critique of the Student's Proof):

The student's proof has an error because they incorrectly assumed the Segment Addition Postulate can be applied in a way that leads to factoring errors.

The critical error is in their substitution and factoring step:

• Incorrect Factoring:
  The student claimed:
  $e^2 + 2ed + d^2 = (e + d)^2$
  This is correct, but their logic in reaching that step from adding areas of triangles seems faulty.

Correct Critique: The similarity properties used and their application of the Segment Addition Postulate are not properly justified.

Do you want a detailed breakdown of why the student’s proof is incorrect? Let me know if you want further clarifications.

Relative Questions for Practice:

1. How do you prove the Pythagorean Theorem using triangle similarity?
2. What is the Segment Addition Postulate, and how is it applied in geometric proofs?
3. How can you identify errors in algebraic proofs involving geometric properties?
4. What are the key properties of similar triangles that are used in proofs?
5. How do you correctly factor algebraic expressions to simplify equations in proofs?

Tip: Always double-check that every step in a proof is mathematically justified, especially when using geometric postulates or similarity rules.