Complete the similar triangles by writing the letters of the triangles in the correct order.

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To correctly complete similar triangles by writing their letters in order, it's essential to match corresponding vertices based on their angles and sides. Here's a step-by-step guide to help you:

### Understanding Similar Triangles

**Similar Triangles** are triangles that have:

1. **Equal corresponding angles**: Each angle in one triangle is equal to the corresponding angle in the other triangle.
2. **Proportional corresponding sides**: The lengths of corresponding sides are in the same proportion.

### Writing the Letters in the Correct Order

When denoting similar triangles, the order of the letters (vertices) should reflect the correspondence between their angles and sides. Here's how to do it:

1. **Identify Corresponding Angles**: Determine which angles in one triangle correspond to which angles in the other triangle.
   
2. **Match the Corresponding Vertices**: The vertices should be ordered so that the first letters correspond to each other, the second letters correspond to each other, and so on.

3. **Use the Similarity Symbol (∼)**: This symbol indicates that the triangles are similar.

### Example

Let's say you have two triangles:

- **Triangle ABC**
- **Triangle DEF**

If ∠A corresponds to ∠D, ∠B to ∠E, and ∠C to ∠F, then the triangles are similar, and you would write:

$$ \triangle ABC \sim \triangle DEF $$

### Steps Illustrated

1. **Identify Correspondence**:
   - Angle A ≈ Angle D
   - Angle B ≈ Angle E
   - Angle C ≈ Angle F

2. **Order the Vertices Accordingly**:
   - First vertices: A and D
   - Second vertices: B and E
   - Third vertices: C and F

3. **Write the Similarity Statement**:
   $$ \triangle ABC \sim \triangle DEF $$

### Additional Tips

- **Consistent Ordering**: Always maintain the same order of corresponding vertices when writing similar triangles.
  
- **Multiple Notations**: If the correspondence is different, adjust the order accordingly. For example, if ∠A corresponds to ∠E, ∠B to ∠D, and ∠C to ∠F, then:
  $$ \triangle ABC \sim \triangle EDF $$

- **Verify Proportions and Angles**: Ensure that both the angles are equal and the sides are proportional to confirm similarity before writing the similarity statement.

### Practice Problem

**Given**:
- Triangle PQR
- Triangle XYZ

If ∠P corresponds to ∠X, ∠Q to ∠Y, and ∠R to ∠Z, write the similarity statement.

**Solution**:
$$
\triangle PQR \sim \triangle XYZ
$$

By following these steps, you can accurately complete and denote similar triangles by writing their vertices in the correct corresponding order.
To complete similar triangles by writing their letters in order, match the corresponding vertices based on equal angles and proportional sides. For example, if ∠A corresponds to ∠D, ∠B to ∠E, and ∠C to ∠F, write the similarity as: $$ \triangle ABC \sim \triangle DEF $$

Complete the similar triangles by writing the letters of the triangles in the correct order.

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