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This image contains three geometry proofs with given statements and diagrams for each proof, along with a two-column format (Statements and Reasons) where students are expected to fill out the steps in each proof.

Here’s an outline to solve each proof:

### Problem 4
**Given:** \( \triangle FGH \) and \( \triangle FJH \) are right triangles, and \( GH \cong HJ \).  
**To Prove:** \( \triangle FGH \cong \triangle FJH \).

**Steps to complete proof:**
1. \( FG \) and \( FJ \) are legs in right triangles, and we know both triangles are right.
2. \( GH \cong HJ \) (given).
3. \( \angle FGH = \angle FJH = 90^\circ \) (definition of a right triangle).
4. By the Hypotenuse-Leg (HL) theorem, \( \triangle FGH \cong \triangle FJH \) (because they share a hypotenuse and have congruent legs).

### Problem 5
**Given:** \( KL \parallel NP \), \( KL \cong NP \).  
**To Prove:** \( \triangle KML \cong \triangle PMN \).

**Steps to complete proof:**
1. \( KL \parallel NP \) (given).
2. \( KL \cong NP \) (given).
3. \( \angle KML \cong \angle PMN \) (alternate interior angles are congruent when lines are parallel).
4. \( KM \cong PN \) (reflexive property as a shared side).
5. By SAS (Side-Angle-Side) criterion, \( \triangle KML \cong \triangle PMN \).

### Problem 6
**Given:** \( TV \parallel XW \), \( \angle TVY \cong \angle XWY \).  
**To Prove:** \( \triangle TVY \cong \triangle XWY \).

**Steps to complete proof:**
1. \( TV \parallel XW \) (given).
2. \( \angle TVY \cong \angle XWY \) (given).
3. \( \angle VYT \cong \angle WYX \) (alternate interior angles are congruent when lines are parallel).
4. \( YV \cong YW \) (reflexive property as a shared side).
5. By ASA (Angle-Side-Angle) criterion, \( \triangle TVY \cong \triangle XWY \).

Would you like a detailed breakdown of any of these proofs, or have questions about specific steps?

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Here are 5 follow-up questions:
1. What is the Hypotenuse-Leg (HL) theorem, and how is it used in right triangle congruence?
2. Why are alternate interior angles important in proving triangle congruence in parallel line setups?
3. How does the reflexive property help in proving congruence in overlapping triangles?
4. What are the criteria (such as SAS, ASA, and HL) for proving triangles congruent?
5. In which situations would the Angle-Angle-Side (AAS) criterion be used instead of the ASA criterion?

**Tip:** Remember, when proving triangles congruent, always look for shared sides or angles and use parallel lines to establish congruent angles through alternate interior or corresponding angle relationships.

Complete the given proofs for each geometry problem shown in the image, with diagrams and statements provided.

Explore detailed solutions to geometry proofs on triangle congruence. Learn how to use the Hypotenuse-Leg, Side-Angle-Side, and Angle-Side-Angle theorems to establish triangle congruence. Suitable for high school students in Grades 9-11.

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