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In this problem, two triangles \( \triangle STU \) and \( \triangle EFG \) are similar, as indicated by the notation \( \triangle STU \sim \triangle EFG \). This implies that the corresponding sides and angles of these triangles are proportional and equal.

### Step-by-step breakdown:

#### 1. **Proportionality of Sides:**
   Since the triangles are similar, we know:
   \[
   \frac{SU}{EF} = \frac{ST}{EG} = \frac{UT}{FG}
   \]
   Using the given sides:
   - \( SU = 25 \), \( ST = 30 \), and \( UT = ? \) for \( \triangle STU \)
   - \( EF = 5 \), \( EG = 9 \), and \( FG = 6 \) for \( \triangle EFG \)

   Let’s calculate the missing side \( UT \):
   \[
   \frac{UT}{FG} = \frac{ST}{EG}
   \]
   Substituting the known values:
   \[
   \frac{UT}{6} = \frac{30}{9}
   \]
   Solving for \( UT \):
   \[
   UT = \frac{30 \times 6}{9} = 20
   \]
   So, \( UT = 20 \).

#### 2. **Finding the Missing Angles:**
   Since the triangles are similar, the corresponding angles are equal. We are given:
   - \( \angle EFG = 82^\circ \)

   Corresponding to this angle, \( \angle STU \) should also be \( 82^\circ \). So, the missing angle \( \angle STU = 82^\circ \).

Thus, the values to be filled in are:
1. \( 82^\circ \) for the angle in \( \triangle STU \)
2. \( 20 \) for the side \( UT \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you calculate the missing angle in a right triangle if two angles are known?
2. What other properties do similar triangles share beyond proportional sides?
3. How do you find the area of two similar triangles if you know the ratio of their sides?
4. Can triangles be similar if their angles are not congruent but their sides are proportional?
5. What are some real-world applications of triangle similarity in architecture or engineering?

**Tip:** When working with similar triangles, always double-check the corresponding sides and angles to ensure proportionality is maintained correctly.

In the given image, find the missing angle and side in two similar triangles, ΔSTU and ΔEFG, by using the proportionality of sides and equality of angles.

This problem involves finding the missing angle and side in two similar triangles ΔSTU and ΔEFG. Learn how to apply the triangle similarity theorem to solve for unknowns using proportionality of sides and equality of angles. Suitable for students in Grades 7-9.

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