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To determine if triangles \( \triangle CDE \) and \( \triangle FGH \) are similar, we need to check if their corresponding angles are congruent or if their corresponding sides are proportional. This is based on the similarity criteria: Angle-Angle (AA), Side-Angle-Side (SAS), or Side-Side-Side (SSS).

### Step-by-Step Analysis:
1. **Angle Comparisons:**
   - In \( \triangle CDE \), two angles are given:
     - \( \angle ECD = 49^\circ \)
     - \( \angle CDE = 82^\circ \)
     Using the triangle sum theorem, we can calculate the third angle \( \angle DCE \):
     \[
     \angle DCE = 180^\circ - 49^\circ - 82^\circ = 49^\circ
     \]
   - In \( \triangle FGH \), one angle is provided:
     - \( \angle HGF = 49^\circ \)
   
   Since \( \angle ECD = \angle HGF \) and both triangles have one angle of \( 49^\circ \), we only need to find one more congruent angle to confirm similarity using the AA criterion.

2. **Side Proportions:**
   - For \( \triangle CDE \):
     - \( CD = 13 \)
     - \( CE = 17 \)
   - For \( \triangle FGH \):
     - \( FG = 39 \)
     - \( GH = 51 \)

   Checking if the sides are proportional:
   \[
   \frac{CD}{FG} = \frac{13}{39} = \frac{1}{3}
   \]
   \[
   \frac{CE}{GH} = \frac{17}{51} = \frac{1}{3}
   \]

Since the corresponding sides are proportional, we can confirm that **the triangles are similar by the Side-Angle-Side (SAS) similarity criterion**, as they have one congruent angle and proportional corresponding sides.

Thus, **the triangles are similar**.

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Would you like a more detailed explanation or have any further questions?

Here are 5 related questions:
1. How do you use the Angle-Angle similarity criterion for triangles?
2. What is the Side-Angle-Side similarity criterion for triangles?
3. How can you calculate the missing angles in a triangle?
4. What is the importance of proportional sides in similar triangles?
5. How does the triangle sum theorem help in solving for unknown angles?

**Tip**: Always check for congruent angles or proportional sides when determining triangle similarity!

Determine if triangle CDE and triangle FGH are or are not similar, and, if they are, state how you know. (Note that figures are NOT necessarily drawn to scale.)

Determine whether triangles CDE and FGH are similar by using the Side-Angle-Side (SAS) criterion. The solution explores angle congruence and side proportions, guiding you through a step-by-step process suitable for Grades 8-10.

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