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To solve the missing angles in each polygon on the worksheet, we can use the formula for the sum of the interior angles of a polygon, which is:

\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]

Where \( n \) is the number of sides of the polygon.

I'll go through each shape one by one:

1. **Quadrilateral (4 sides):**
   - Known angles: \( 85^\circ, 110^\circ, 85^\circ \)
   - Sum of interior angles: \( (4 - 2) \times 180^\circ = 360^\circ \)
   - Missing angle \( x \):
   \[
   x = 360^\circ - (85^\circ + 110^\circ + 85^\circ) = 80^\circ
   \]

2. **Quadrilateral (4 sides):**
   - Known angles: \( 90^\circ, 97^\circ, 90^\circ \)
   - Sum of interior angles: \( 360^\circ \)
   - Missing angle \( x \):
   \[
   x = 360^\circ - (90^\circ + 97^\circ + 90^\circ) = 83^\circ
   \]

3. **Triangle (3 sides):**
   - Known angles: \( 57^\circ, 57^\circ \)
   - Sum of interior angles: \( (3 - 2) \times 180^\circ = 180^\circ \)
   - Missing angle \( x \):
   \[
   x = 180^\circ - (57^\circ + 57^\circ) = 66^\circ
   \]

4. **Hexagon (6 sides):**
   - Known angles: \( 128^\circ, 130^\circ, 108^\circ, 119^\circ, 104^\circ \)
   - Sum of interior angles: \( (6 - 2) \times 180^\circ = 720^\circ \)
   - Missing angle \( x \):
   \[
   x = 720^\circ - (128^\circ + 130^\circ + 108^\circ + 119^\circ + 104^\circ) = 131^\circ
   \]

5. **Quadrilateral (4 sides):**
   - Known angles: \( 123^\circ, 120^\circ, 93^\circ \)
   - Sum of interior angles: \( 360^\circ \)
   - Missing angle \( x \):
   \[
   x = 360^\circ - (123^\circ + 120^\circ + 93^\circ) = 24^\circ
   \]

6. **Pentagon (5 sides):**
   - Known angles: \( 118^\circ, 115^\circ, 127^\circ, 102^\circ \)
   - Sum of interior angles: \( (5 - 2) \times 180^\circ = 540^\circ \)
   - Missing angle \( x \):
   \[
   x = 540^\circ - (118^\circ + 115^\circ + 127^\circ + 102^\circ) = 78^\circ
   \]

7. **Quadrilateral (4 sides):**
   - Known angles: \( 128^\circ, 128^\circ, 107^\circ \)
   - Sum of interior angles: \( 360^\circ \)
   - Missing angle \( x \):
   \[
   x = 360^\circ - (128^\circ + 128^\circ + 107^\circ) = 87^\circ
   \]

Would you like to go through any particular problem in more detail, or do you have any questions?

### Relative Questions:
1. What is the formula for calculating the sum of the interior angles of any polygon?
2. How does the number of sides in a polygon affect the sum of its interior angles?
3. Why is the sum of the interior angles of a triangle always 180°?
4. Can a polygon have an interior angle sum that is not a multiple of 180°? Why or why not?
5. How would you find the measure of an exterior angle if the interior angles are known?

### Tip:
Always double-check the sum of given angles before subtracting to ensure accuracy when finding the missing angle.

Solve a variety of polygon angle problems using the Interior Angle Sum Theorem. This worksheet covers hexagons, pentagons, triangles, and quadrilaterals, helping students learn how to find missing angles. Suitable for Grades 6-8.

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