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Let's go through each problem in detail.

### Top Section: Congruency Statements (SSS or SAS)

1. **Problem 1:**
   - **Given:** Triangles with sides marked on \( \triangle WXY \) and \( \triangle FPC \).
   - **Solution:** These triangles can be proven congruent by the **SSS (Side-Side-Side)** criterion, as all three pairs of corresponding sides are marked congruent.
   - **Congruency Statement:** \( \triangle WXY \cong \triangle FPC \) by SSS.

2. **Problem 2:**
   - **Given:** Triangles \( \triangle ABD \) and \( \triangle CEB \) with two pairs of congruent sides and a shared side \( BE \).
   - **Solution:** These triangles can be proven congruent by **SAS (Side-Angle-Side)**, as two pairs of sides and the included angle \( \angle ABE = \angle CBE \) are congruent.
   - **Congruency Statement:** \( \triangle ABD \cong \triangle CEB \) by SAS.

3. **Problem 3:**
   - **Given:** Triangles \( \triangle TUY \) and \( \triangle VUY \) with two sides and a shared angle.
   - **Solution:** Congruent by **SAS**, with two pairs of sides and the included angle \( \angle TUY = \angle VUY \).
   - **Congruency Statement:** \( \triangle TUY \cong \triangle VUY \) by SAS.

4. **Problem 4:**
   - **Given:** Quadrilateral with two overlapping triangles \( \triangle MNQ \) and \( \triangle PNQ \).
   - **Solution:** The triangles are congruent by **SSS**, as all three corresponding sides are congruent.
   - **Congruency Statement:** \( \triangle MNQ \cong \triangle PNQ \) by SSS.

5. **Problem 5:**
   - **Given:** \( \triangle JLK \) and \( \triangle KJL \) with three pairs of congruent sides.
   - **Solution:** Congruent by **SSS**, as all pairs of sides are equal.
   - **Congruency Statement:** \( \triangle JLK \cong \triangle KJL \) by SSS.

6. **Problem 6:**
   - **Given:** \( \triangle WXZ \) and \( \triangle YXZ \) with two sides and a shared angle.
   - **Solution:** Congruent by **SAS** with two pairs of sides and the included angle.
   - **Congruency Statement:** \( \triangle WXZ \cong \triangle YXZ \) by SAS.

### Bottom Section: Determine Congruency (SSS, SAS, ASA, AAS, HL, or Not Congruent)

1. **Problem 1:**
   - **Solution:** These triangles appear congruent by **SAS**, given two sides and the included angle.
   - **Answer:** SAS.

2. **Problem 2:**
   - **Solution:** Not enough information to determine congruence, as we lack both side and angle measurements.
   - **Answer:** Not congruent.

3. **Problem 3:**
   - **Solution:** Congruent by **SSS**, with three pairs of corresponding sides congruent.
   - **Answer:** SSS.

4. **Problem 4:**
   - **Solution:** Not enough information to determine congruence.
   - **Answer:** Not congruent.

5. **Problem 5:**
   - **Solution:** Congruent by **HL (Hypotenuse-Leg)**, as the triangles are right triangles with a pair of legs and hypotenuses congruent.
   - **Answer:** HL.

6. **Problem 6:**
   - **Solution:** Not enough information to conclude congruence.
   - **Answer:** Not congruent.

Would you like further details on any of these solutions or have questions on a specific problem?

### Additional Related Questions

1. What other criteria can determine triangle congruency beyond SSS and SAS?
2. How can ASA and AAS be used in congruency proofs?
3. In what situations can the Hypotenuse-Leg (HL) theorem be applied?
4. How do we know which pairs of sides or angles are congruent in a diagram?
5. What makes a pair of triangles "not congruent" in the absence of information?

#### Tip
Always verify which sides or angles are marked congruent in diagrams to choose the right congruency criterion.

State whether the triangles could be proven congruent, if possible, by SSS or SAS. Include a congruency statement for all congruent triangles. Determine if the triangles can be proven congruent, if possible, by SSS, SAS, ASA, AAS, or HL.

Learn how to determine triangle congruency using various criteria, including SSS, SAS, ASA, AAS, and HL. This guide provides step-by-step solutions for identifying congruent triangles and writing congruency statements. Suitable for students in Grades 8-10.

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