Based on the information given below, which two of the triangles below must be congruent? Enter your answer as a list of numbers. If you think the first and second triangles are congruent, enter "1,2".

[asy]
size(15cm);
pair A[]; 
pair B[]; 
pair C[]; 
pair nudge= (13,0); 
A[0] = (0,0); 
B[0] = (5,9); 
C[0] = (8,1);
for(int i =1; i< 5; ++i){
  A[i] = shift(nudge)*A[i-1];
  B[i] = shift(nudge)*B[i-1]; 
  C[i] = shift(nudge)*C[i-1];
}
for(int i =0; i<5; ++i){ 
   draw(A[i]--B[i]--C[i]--cycle);
   label("$13$", (A[i]+B[i])/2, W); 
   label("$12$", (B[i]+C[i])/2, E); 
}
label(scale(0.75)*"$62^{\circ}$", A[0]+0.65*dir(35), dir(35)); 
label(scale(0.75)*"$42^{\circ}$", C[1]+0.25*dir(145), dir(145)); 
label(scale(0.75)*"$40^{\circ}$", A[2]+0.65*dir(35), dir(35)); 
label(scale(0.75)*"$44^{\circ}$", C[2]+0.25*dir(145), dir(145)); 
label(scale(0.75)*"$45^{\circ}$", B[3]+1.25*dir(-100), S); 
label(scale(0.75)*"$73^{\circ}$", C[4]+0.25*dir(145), dir(145)); 
label(scale(0.75)*"$62^{\circ}$", A[4]+0.65*dir(35), dir(35)); 
[/asy]

To determine which triangles are congruent, we need to compare the given information for each triangle. Specifically, we should look for triangles that share the same side lengths and angles. From the diagram:

- **Side lengths**: Every triangle has sides labeled as 13 and 12.
- **Angles**: The angles given for each triangle are as follows:
  1. First triangle: $62^\circ$
  2. Second triangle: $42^\circ$
  3. Third triangle: $40^\circ$ and $44^\circ$
  4. Fourth triangle: $45^\circ$
  5. Fifth triangle: $62^\circ$ and $73^\circ$

### Step-by-step analysis:
- Triangles with the same angles and side lengths will be congruent. 
- The angles provided for the first and fifth triangles are both $62^\circ$. Since both triangles also have sides labeled as 13 and 12, it is likely that these two triangles are congruent.

Thus, the **first** and **fifth** triangles are congruent.

### Final answer:
The two congruent triangles are **1,5**.

---

Would you like more details on how to verify congruency using the Angle-Side-Angle (ASA) or another method?

Here are five related questions to extend this understanding:
1. How does the Angle-Side-Angle (ASA) theorem prove triangle congruence?
2. Can triangles with only equal sides but different angles still be congruent?
3. What other congruence criteria can be applied to triangles?
4. How would side lengths of other triangles affect their potential congruence?
5. Could two triangles with the same angles but different side lengths ever be congruent?

**Tip**: To confirm congruence, always ensure that at least two angles and the included side, or two sides and the included angle, match between triangles.

Solve a triangle congruence problem where you identify which triangles are congruent based on given side lengths and angles. This problem uses geometry principles like the Angle-Side-Angle (ASA) theorem. Suitable for students in Grades 8-10.

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