Find missing interior and exterior angles in various polygons on this math worksheet.

To solve the problems involving the interior and exterior angles of polygons, we need to use the following key properties:

1. Sum of Interior Angles of a Polygon: For an \( n \)-sided polygon, the sum of the interior angles is given by:
\[
(n-2) \times 180^\circ
\]

2. Exterior Angle Property: The exterior angle of a polygon is supplementary to its corresponding interior angle. That is, if an interior angle is \( \theta \), then the exterior angle is \( 180^\circ - \theta \).

3. Sum of Angles in a Triangle: The sum of the interior angles in a triangle is always \( 180^\circ \).

4. Sum of Angles on a Straight Line: The sum of angles on a straight line is \( 180^\circ \).

Let's solve each problem step by step.

---

Problem 1

The polygon is a quadrilateral. The sum of the interior angles of a quadrilateral is:
\[
(4-2) \times 180^\circ = 360^\circ
\]
Given angles are \( 84^\circ \), \( 100^\circ \), \( 121^\circ \), and \( x^\circ \). We can set up the equation:
\[
84^\circ + 100^\circ + 121^\circ + x^\circ = 360^\circ
\]
Simplify:
\[
305^\circ + x^\circ = 360^\circ
\]
Solve for \( x \):
\[
x = 360^\circ - 305^\circ = 55^\circ
\]
So, \( x = 55^\circ \).

---

Problem 2

The polygon is a triangle. The sum of the interior angles of a triangle is \( 180^\circ \). Given angles are \( x^\circ \), \( 100^\circ \), and \( 120^\circ \). We can set up the equation:
\[
x^\circ + 100^\circ + 120^\circ = 180^\circ
\]
Simplify:
\[
x^\circ + 220^\circ = 180^\circ
\]
Solve for \( x \):
\[
x = 180^\circ - 220^\circ = -40^\circ
\]
This result is incorrect because it suggests a negative angle, which is not possible. Let's recheck the problem: The given angles \( 100^\circ \) and \( 120^\circ \) cannot both be interior angles of a triangle since their sum exceeds \( 180^\circ \). It seems there might be a misunderstanding. Assuming the problem meant to find the exterior angle, we can proceed as follows:
The exterior angle at the vertex with \( x^\circ \) is:
\[
x^\circ = 180^\circ - 100^\circ = 80^\circ
\]
So, \( x = 80^\circ \).

---

Problem 3

The polygon is a quadrilateral. The sum of the interior angles of a quadrilateral is \( 360^\circ \). Given angles are \( 95^\circ \), \( 70^\circ \), \( 90^\circ \) (right angle), and \( x^\circ \). We can set up the equation:
\[
95^\circ + 70^\circ + 90^\circ + x^\circ = 360^\circ
\]
Simplify:
\[
255^\circ + x^\circ = 360^\circ
\]
Solve for \( x \):
\[
x = 360^\circ - 255^\circ = 105^\circ
\]
So, \( x = 105^\circ \).

---

Problem 4

The polygon is a quadrilateral. The sum of the interior angles of a quadrilateral is \( 360^\circ \). Given angles are \( 40^\circ \), \( 60^\circ \), \( 70^\circ \), and \( 70^\circ \), and the unknown angle is \( x^\circ \). We can set up the equation:
\[
40^\circ + 60^\circ + 70^\circ + 70^\circ + x^\circ = 360^\circ
\]
Simplify:
\[
240^\circ + x^\circ = 360^\circ
\]
Solve for \( x \):
\[
x = 360^\circ - 240^\circ = 120^\circ
\]
So, \( x = 120^\circ \).

---

Problem 5

The polygon is a quadrilateral. The sum of the interior angles of a quadrilateral is \( 360^\circ \). Given angles are \( 55^\circ \), \( 93^\circ \), \( 102^\circ \), and \( x^\circ \). We can set up the equation:
\[
55^\circ + 93^\circ + 102^\circ + x^\circ = 360^\circ
\]
Simplify:
\[
250^\circ + x^\circ = 360^\circ
\]
Solve for \( x \):
\[
x = 360^\circ - 250^\circ = 110^\circ
\]
So, \( x = 110^\circ \).

---

Problem 6

The polygon is a triangle. The sum of the interior angles of a triangle is \( 180^\circ \). Given angles are \( 31^\circ \), \( 91^\circ \), and \( x^\circ \). We can set up the equation:
\[
31^\circ + 91^\circ + x^\circ = 180^\circ
\]
Simplify:
\[
122^\circ + x^\circ = 180^\circ
\]
Solve for \( x \):
\[
x = 180^\circ - 122^\circ = 58^\circ
\]
So, \( x = 58^\circ \).

---

Problem 7

The polygon is a pentagon. The sum of the interior angles of a pentagon is:
\[
(5-2) \times 180^\circ = 540^\circ
\]
Given angles are \( 90^\circ \), \( 168^\circ \), \( 150^\circ \), \( 63^\circ \), and \( x^\circ \). We can set up the equation:
\[
90^\circ + 168^\circ + 150^\circ + 63^\circ + x^\circ = 540^\circ
\]
Simplify:
\[
471^\circ + x^\circ = 540^\circ
\]
Solve for \( x \):
\[
x = 540^\circ - 471^\circ = 69^\circ
\]
So, \( x = 69^\circ \).

---

Problem 8

The polygon is a triangle. The sum of the interior angles of a triangle is \( 180^\circ \). Given angles are \( 45^\circ \), \( 80^\circ \), and \( x^\circ \). We can set up the equation:
\[
45^\circ + 80^\circ + x^\circ = 180^\circ
\]
Simplify:
\[
125^\circ + x^\circ = 180^\circ
\]
Solve for \( x \):
\[
x = 180^\circ - 125^\circ = 55^\circ
\]
So, \( x = 55^\circ \).

---

Final Answers

\[
\boxed{55, 80, 105, 120, 110, 58, 69, 55}
\]