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To solve these problems, we need to calculate the missing interior angles, denoted by \( x \), using properties of polygons and the fact that the sum of interior angles in any polygon can be determined by the formula:

\[
S = 180(n-2)
\]
Where:
- \( S \) is the sum of the interior angles.
- \( n \) is the number of sides of the polygon.

Let's solve the problems one by one.

### 1. Trapezoid (4 sides)
- Given angles: \( 135^\circ, 60^\circ, 50^\circ \).
- The sum of the interior angles of a quadrilateral is \( 360^\circ \).
- Equation: \( 135^\circ + 60^\circ + 50^\circ + x = 360^\circ \)
  \[
  x = 360^\circ - (135^\circ + 60^\circ + 50^\circ) = 360^\circ - 245^\circ = 115^\circ
  \]
  **\( x = 115^\circ \)**

### 2. Quadrilateral (4 sides)
- Given angles: \( 90^\circ, 90^\circ, 90^\circ \).
- Again, the sum is \( 360^\circ \).
- Equation: \( 90^\circ + 90^\circ + 90^\circ + x = 360^\circ \)
  \[
  x = 360^\circ - 270^\circ = 90^\circ
  \]
  **\( x = 90^\circ \)**

### 3. Triangle (3 sides)
- Given angles: \( 35^\circ, 60^\circ \).
- The sum of the interior angles of a triangle is \( 180^\circ \).
- Equation: \( 35^\circ + 60^\circ + x = 180^\circ \)
  \[
  x = 180^\circ - (35^\circ + 60^\circ) = 180^\circ - 95^\circ = 85^\circ
  \]
  **\( x = 85^\circ \)**

### 4. Hexagon (6 sides)
- Given angles: \( 122^\circ, 130^\circ, 130^\circ, 138^\circ, 128^\circ \).
- The sum of the interior angles of a hexagon is \( 720^\circ \) (using the formula \( 180(6-2) \)).
- Equation: \( 122^\circ + 130^\circ + 130^\circ + 138^\circ + 128^\circ + x = 720^\circ \)
  \[
  x = 720^\circ - (122^\circ + 130^\circ + 130^\circ + 138^\circ + 128^\circ) = 720^\circ - 648^\circ = 72^\circ
  \]
  **\( x = 72^\circ \)**

### 5. Quadrilateral (4 sides)
- Given angles: \( 120^\circ, 120^\circ, 107^\circ \).
- The sum is \( 360^\circ \).
- Equation: \( 120^\circ + 120^\circ + 107^\circ + x = 360^\circ \)
  \[
  x = 360^\circ - (120^\circ + 120^\circ + 107^\circ) = 360^\circ - 347^\circ = 13^\circ
  \]
  **\( x = 13^\circ \)**

### 6. Quadrilateral (4 sides)
- Given angles: \( 70^\circ, 90^\circ \).
- Sum is \( 360^\circ \).
- Equation: \( 70^\circ + 90^\circ + 90^\circ + x = 360^\circ \)
  \[
  x = 360^\circ - (70^\circ + 90^\circ + 90^\circ) = 360^\circ - 250^\circ = 110^\circ
  \]
  **\( x = 110^\circ \)**

### 7. Quadrilateral (4 sides)
- Given angles: \( 60^\circ, 60^\circ, 120^\circ \).
- Sum is \( 360^\circ \).
- Equation: \( 60^\circ + 60^\circ + 120^\circ + x = 360^\circ \)
  \[
  x = 360^\circ - (60^\circ + 60^\circ + 120^\circ) = 360^\circ - 240^\circ = 120^\circ
  \]
  **\( x = 120^\circ \)**

### 8. Rhombus (4 sides)
- Given angles: \( 140^\circ, 40^\circ, 140^\circ \).
- The sum is \( 360^\circ \).
- Equation: \( 140^\circ + 40^\circ + 140^\circ + x = 360^\circ \)
  \[
  x = 360^\circ - (140^\circ + 40^\circ + 140^\circ) = 360^\circ - 320^\circ = 40^\circ
  \]
  **\( x = 40^\circ \)**

---

Do you have any questions or need more details on any of the problems? Here are five related questions to expand on this topic:

1. How do you calculate the sum of interior angles for polygons with more than 4 sides?
2. What is the relationship between the number of sides of a polygon and the measure of its interior angles?
3. How do exterior angles of a polygon relate to its interior angles?
4. Can you solve for interior angles of irregular polygons?
5. How would you calculate the measure of each angle in a regular polygon?

**Tip:** Always double-check your calculations to ensure that the sum of the angles matches the expected total for that polygon.

Find the missing interior angles of the given polygons.

Learn how to find the missing interior angles in polygons with this detailed guide. Using the interior angle sum formula and basic angle properties, we solve problems for quadrilaterals, triangles, hexagons, and rhombuses. Suitable for middle school students.

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