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.

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!