Here’s a breakdown of the problems in the worksheet and solutions for each:

Part 1: Completing Congruence Statements for $\triangle ABC \cong \triangle YZX$

1. $\angle X \cong$ $\angle A$
2. $AB \cong$ $YZ$
3. $\angle C \cong$ $\angle Z$
4. $\angle Y \cong$ $\angle B$
5. $\angle Z \cong$ $\angle C$
6. $ZX \cong$ $AC$

Part 2: Completing Statements for $\triangle ART \cong \triangle CAN$

7. $\angle R \cong$ $\angle A$
8. $CA \cong$ $CA$ (shared side)
9. $\angle C \cong$ $\angle C$
10. $\angle T \cong$ $\angle N$
11. $TR \cong$ $AN$
12. $CN \cong$ $RT$

Part 3: $\triangle ABC \cong \triangle DEF$

13. $BC \cong$ $EF$
14. Solve for $x$:
    $3x = 9 \rightarrow x = 3$
15. Solve for $m\angle B$:
    $m\angle B = 87^\circ$
16. $\triangle ABC \cong \triangle DEF$ by $SAS$ (Side-Angle-Side).
17. $\angle A \cong \angle D$
18. Solve for $y$:
    $5x + 2 = 27 \rightarrow x = 5$, then $y = 5x = 25$.
19. $m\angle C = 42^\circ$
20. $\triangle CAB \cong \triangle EFD$ by $ASA$ (Angle-Side-Angle).

Part 4: Additional Information Needed for Congruence

21. To prove the triangles congruent, we need $\angle FDE \cong \angle ABC$ (Angle-Angle-Side).
22. To prove the triangles congruent, we need $DE \cong EF$ (Side-Side-Side or SAS).

Would you like further clarification on any section or more explanation for solving specific parts?

Here are 5 questions to expand your understanding:

1. Why is $SAS$ a valid postulate for proving triangle congruence?
2. How can shared sides between two triangles simplify congruence proofs?
3. Why is it important to solve for $x$ and $y$ to complete congruence statements?
4. What are the key differences between $ASA$ and $AAS$ postulates?
5. How can you use the properties of angles to confirm triangle congruence?

Tip: Always double-check that the triangles meet all the conditions of the congruence postulate or theorem you're applying.