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.