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.