To solve this problem, let's analyze the information given:

1. Similarity of Triangles: The two triangles $\triangle ABC$ and $\triangle EFG$ are similar, which means their corresponding angles are equal, and their corresponding side lengths are proportional.

2. Angle Measures: In $\triangle ABC$, angle $\angle A$ is given as $42^\circ$. Since the triangles are similar, the corresponding angle $\angle E$ in $\triangle EFG$ must also be $42^\circ$.

3. Side Length Ratios: Since the triangles are similar, we can set up a proportion between the corresponding side lengths. We are given:

   • $AB = 221$
   • $EF = 195$

   We can use these lengths to find the scale factor (ratio) between $\triangle ABC$ and $\triangle EFG$: $\frac{AB}{EF} = \frac{221}{195}$

4. Finding Length of Side: If the question asks for a specific side in $\triangle EFG$, we can use this ratio to find the missing length. For example, if it asks for side $EG$ (corresponding to $AC = 119$), we can write: $\frac{AC}{EG} = \frac{AB}{EF}$ Solving for $EG$: $EG = \frac{AC \cdot EF}{AB} = \frac{119 \cdot 195}{221}$

Let's calculate this value:

$EG = \frac{119 \cdot 195}{221} \approx 105.14$

Answers

• Measure of angle $E$: $42^\circ$.
• Length of side $EG$: approximately $105.14$.

Would you like more details on any step, or do you have further questions?

Here are some related questions to deepen understanding:

1. How do we know which sides correspond in similar triangles?
2. Can we solve for any angle or side length if we only know one side length and one angle?
3. How would this process change if the triangles were not similar?
4. What if both triangles were right triangles—how might that simplify solving for unknowns?
5. How could we use trigonometry if additional angles were given?

Tip: In similar triangles, always match corresponding angles and sides to set up correct proportions!