To determine whether triangles $\triangle CDE$ and $\triangle FGH$ are similar, we check if their corresponding angles are equal or their sides are proportional. Triangles are similar if they meet either the AA (Angle-Angle) similarity criterion or the SSS (Side-Side-Side) or SAS (Side-Angle-Side) proportionality criteria.

Given Information:

• For $\triangle CDE$:

  • $\angle CDE = 50^\circ$,
  • Side lengths: $CD = 20$, $DE = 28$.

• For $\triangle FGH$:

  • $\angle GFH = 50^\circ$, $\angle FGH = 82^\circ$,
  • Side lengths: $GF = 60$, $FH = 81$, $GH = 63$.

Step 1: Check angles for $\triangle CDE$ and $\triangle FGH$

1. Both triangles have $\angle CDE = \angle GFH = 50^\circ$.
2. $\triangle FGH$ has $\angle FGH = 82^\circ$, which makes the third angle $\angle FGH = 180^\circ - (82^\circ + 50^\circ) = 48^\circ$.
3. $\triangle CDE$ has $\angle CDE = 50^\circ$, so the remaining angle is $180^\circ - (50^\circ + 50^\circ) = 80^\circ$.

Step 2: Compare side proportions

We calculate the side ratios for corresponding sides:

1. $\frac{CD}{GF} = \frac{20}{60} = \frac{1}{3}$,
2. $\frac{DE}{FH} = \frac{28}{81} \approx 0.345$,
3. $\frac{CE}{GH}$ (needs to be determined from the triangle).

The sides are not proportional.

Conclusion:

The triangles are not similar because they fail the angle similarity criterion (AA) and the sides are not proportional (SSS or SAS). Let me know if you'd like a more detailed explanation!

Would you like to review other similarity criteria or discuss this further?