Let's analyze and solve the missing angles in the provided polygons. Here's the breakdown for each problem:

Problem 1:

• The given angles are $84^\circ$, $100^\circ$, and $121^\circ$.

• The sum of the interior angles of a quadrilateral is $360^\circ$.

  $x = 360^\circ - (84^\circ + 100^\circ + 121^\circ) = 360^\circ - 305^\circ = 55^\circ$

  So, $x = 55^\circ$.

Problem 2:

• The given angles are $100^\circ$ and $120^\circ$.

• For a triangle, the sum of the interior angles is $180^\circ$.

  $x = 180^\circ - (100^\circ + 120^\circ) = 180^\circ - 220^\circ = -40^\circ$

  But this cannot be correct since a triangle cannot have a negative angle, so the problem might have an error. Please verify the image again for this one.

Problem 3:

• The given angles are $95^\circ$ and $70^\circ$.

• The sum of the interior angles of a triangle is $180^\circ$.

  $x = 180^\circ - (95^\circ + 70^\circ) = 180^\circ - 165^\circ = 15^\circ$

  So, $x = 15^\circ$.

Problem 4:

• The given angles are $120^\circ$, $85^\circ$, and $70^\circ$.

• This is a quadrilateral, and the sum of interior angles is $360^\circ$.

  $x = 360^\circ - (120^\circ + 85^\circ + 70^\circ) = 360^\circ - 275^\circ = 85^\circ$

  So, $x = 85^\circ$.

Problem 5:

• The given angles are $55^\circ$, $102^\circ$, and $80^\circ$.

• This is a quadrilateral.

  $x = 360^\circ - (55^\circ + 102^\circ + 80^\circ) = 360^\circ - 237^\circ = 123^\circ$

  So, $x = 123^\circ$.

Problem 6:

• The given angles are $91^\circ$ and $80^\circ$.

• The sum of the interior angles of a triangle is $180^\circ$.

  $x = 180^\circ - (91^\circ + 80^\circ) = 180^\circ - 171^\circ = 9^\circ$

  So, $x = 9^\circ$.

Problem 7:

• The given angles are $75^\circ$, $45^\circ$, and $63^\circ$.

• The sum of the interior angles of a quadrilateral is $360^\circ$.

  $x = 360^\circ - (75^\circ + 45^\circ + 63^\circ) = 360^\circ - 183^\circ = 177^\circ$

  So, $x = 177^\circ$.

Problem 8:

• The given angles are $80^\circ$ and $63^\circ$.

• This is a triangle.

  $x = 180^\circ - (80^\circ + 63^\circ) = 180^\circ - 143^\circ = 37^\circ$

  So, $x = 37^\circ$.

Would you like further details on any of the steps?

5 Related Questions:

1. How do you calculate the sum of interior angles for any polygon?
2. What are the exterior angles of a polygon, and how are they related to the interior angles?
3. How do you handle problems with missing angles in polygons?
4. Can the sum of exterior angles of any polygon change with the number of sides?
5. What is the role of supplementary angles in finding missing angles in polygons?

Tip:

Always check if the sum of angles provided makes sense for the polygon in question (e.g., the sum of the interior angles of a triangle must be 180°).