I can assist with the contents of the uploaded image. The worksheet focuses on proving triangle similarity using different methods (AA~, SSS, etc.). Here's how we can analyze and solve this:

Breakdown of the Problems:

1. Problem 1:
   Given: $DE \parallel GH$
   Prove: $\triangle DFE \sim \triangle GFH$
   Reasoning:

   • $DE \parallel GH$ (Given)
   • Angles $\angle DFE$ and $\angle GFH$ are congruent due to the Alternate Interior Angles Theorem.
   • Similarly, $\angle DEF$ and $\angle FGH$ are congruent (Alternate Interior Angles Theorem).
   • Therefore, by AA~ (Angle-Angle similarity), $\triangle DFE \sim \triangle GFH$.

2. Problem 2:
   Given: $\frac{QS}{TV} = \frac{RS}{UV} = 3$, $QR = 3TV$
   Prove: $\triangle QRS \sim \triangle TUV$
   Reasoning:

   • The side ratios $\frac{QS}{TV}$, $\frac{RS}{UV}$, and $\frac{QR}{TU}$ are equal, which satisfies the SSS (Side-Side-Side) similarity criterion.
   • Therefore, $\triangle QRS \sim \triangle TUV$.

3. Problem 3:
   Given: $\frac{LM}{LJ} = \frac{LN}{LK}$
   Prove: $\triangle LMN \sim \triangle LJK$
   Reasoning:

   • The given proportional sides imply that the two triangles share equal ratios.
   • Additionally, $\angle L$ is congruent in both triangles (common angle).
   • Therefore, $\triangle LMN \sim \triangle LJK$ by SAS~ (Side-Angle-Side similarity).

Would you like detailed written steps or clarifications for any specific proof?

Expanding Knowledge:

1. What is the difference between AA~, SAS~, and SSS in triangle similarity proofs?
2. How does the concept of parallel lines help in proving triangle similarity?
3. Can you think of a real-life application where similar triangles are useful?
4. Why is it essential to understand proportionality in geometry?
5. How can the Alternate Interior Angles Theorem be applied to other problems involving parallel lines?

Tip: Always mark the given information and key angles on your diagram to simplify proofs and avoid missing relationships!