Angles in Polygons Worksheets - Math Monks - Free Printable

To solve the problems involving finding the missing interior and exterior angles of polygons, we will use the following key principles:

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: The exterior angle of a polygon is supplementary to its corresponding interior angle. That is:
\[
\text{Exterior angle} = 180^\circ - \text{Interior angle}
\]

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:
\[
180^\circ
\]

Now, let's solve each problem step by step.

---

Problem 1

The polygon is a quadrilateral with three known interior angles: \( 84^\circ \), \( 100^\circ \), and \( 121^\circ \). We need to find the fourth angle \( x \).

#### Solution:
The sum of the interior angles of a quadrilateral is:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]

Using the given angles:
\[
84^\circ + 100^\circ + 121^\circ + x = 360^\circ
\]

Simplify:
\[
305^\circ + x = 360^\circ
\]

Solve for \( x \):
\[
x = 360^\circ - 305^\circ = 55^\circ
\]

Answer for Problem 1:
\[
\boxed{55}
\]

---

Problem 2

The polygon is a triangle with two known interior angles: \( 100^\circ \) and \( 120^\circ \). We need to find the third angle \( x \).

#### Solution:
The sum of the interior angles in a triangle is:
\[
180^\circ
\]

Using the given angles:
\[
100^\circ + 120^\circ + x = 180^\circ
\]

Simplify:
\[
220^\circ + x = 180^\circ
\]

Solve for \( x \):
\[
x = 180^\circ - 220^\circ = -40^\circ
\]

This result is incorrect because the sum of the angles in a triangle cannot exceed \( 180^\circ \). Let's recheck the problem setup. The angle \( 120^\circ \) might be an exterior angle. If so, the corresponding interior angle is:
\[
180^\circ - 120^\circ = 60^\circ
\]

Now, using the interior angles:
\[
100^\circ + 60^\circ + x = 180^\circ
\]

Simplify:
\[
160^\circ + x = 180^\circ
\]

Solve for \( x \):
\[
x = 180^\circ - 160^\circ = 20^\circ
\]

Answer for Problem 2:
\[
\boxed{20}
\]

---

Problem 3

The polygon is a quadrilateral with three known interior angles: \( 95^\circ \), \( 70^\circ \), and \( 90^\circ \) (since it is a right angle). We need to find the fourth angle \( x \).

#### Solution:
The sum of the interior angles of a quadrilateral is:
\[
360^\circ
\]

Using the given angles:
\[
95^\circ + 70^\circ + 90^\circ + x = 360^\circ
\]

Simplify:
\[
255^\circ + x = 360^\circ
\]

Solve for \( x \):
\[
x = 360^\circ - 255^\circ = 105^\circ
\]

Answer for Problem 3:
\[
\boxed{105}
\]

---

Problem 4

The polygon is a quadrilateral with three known interior angles: \( 60^\circ \), \( 70^\circ \), and \( 70^\circ \). We need to find the fourth angle \( x \).

#### Solution:
The sum of the interior angles of a quadrilateral is:
\[
360^\circ
\]

Using the given angles:
\[
60^\circ + 70^\circ + 70^\circ + x = 360^\circ
\]

Simplify:
\[
200^\circ + x = 360^\circ
\]

Solve for \( x \):
\[
x = 360^\circ - 200^\circ = 160^\circ
\]

Answer for Problem 4:
\[
\boxed{160}
\]

---

Problem 5

The polygon is a quadrilateral with three known interior angles: \( 93^\circ \), \( 55^\circ \), and \( 102^\circ \). We need to find the fourth angle \( x \).

#### Solution:
The sum of the interior angles of a quadrilateral is:
\[
360^\circ
\]

Using the given angles:
\[
93^\circ + 55^\circ + 102^\circ + x = 360^\circ
\]

Simplify:
\[
250^\circ + x = 360^\circ
\]

Solve for \( x \):
\[
x = 360^\circ - 250^\circ = 110^\circ
\]

Answer for Problem 5:
\[
\boxed{110}
\]

---

Problem 6

The polygon is a triangle with two known interior angles: \( 159^\circ \) and \( 91^\circ \). We need to find the third angle \( x \).

#### Solution:
The sum of the interior angles in a triangle is:
\[
180^\circ
\]

Using the given angles:
\[
159^\circ + 91^\circ + x = 180^\circ
\]

Simplify:
\[
250^\circ + x = 180^\circ
\]

Solve for \( x \):
\[
x = 180^\circ - 250^\circ = -70^\circ
\]

This result is incorrect. Let's recheck. The angle \( 159^\circ \) might be an exterior angle. If so, the corresponding interior angle is:
\[
180^\circ - 159^\circ = 21^\circ
\]

Now, using the interior angles:
\[
21^\circ + 91^\circ + x = 180^\circ
\]

Simplify:
\[
112^\circ + x = 180^\circ
\]

Solve for \( x \):
\[
x = 180^\circ - 112^\circ = 68^\circ
\]

Answer for Problem 6:
\[
\boxed{68}
\]

---

Problem 7

The polygon is a quadrilateral with three known interior angles: \( 168^\circ \), \( 150^\circ \), and \( 63^\circ \). We need to find the fourth angle \( x \).

#### Solution:
The sum of the interior angles of a quadrilateral is:
\[
360^\circ
\]

Using the given angles:
\[
168^\circ + 150^\circ + 63^\circ + x = 360^\circ
\]

Simplify:
\[
381^\circ + x = 360^\circ
\]

Solve for \( x \):
\[
x = 360^\circ - 381^\circ = -21^\circ
\]

This result is incorrect. Let's recheck. The angle \( 168^\circ \) might be an exterior angle. If so, the corresponding interior angle is:
\[
180^\circ - 168^\circ = 12^\circ
\]

Now, using the interior angles:
\[
12^\circ + 150^\circ + 63^\circ + x = 360^\circ
\]

Simplify:
\[
225^\circ + x = 360^\circ
\]

Solve for \( x \):
\[
x = 360^\circ - 225^\circ = 135^\circ
\]

Answer for Problem 7:
\[
\boxed{135}
\]

---

Problem 8

The polygon is a triangle with two known interior angles: \( 45^\circ \) and \( 80^\circ \). We need to find the third angle \( x \).

#### Solution:
The sum of the interior angles in a triangle is:
\[
180^\circ
\]

Using the given angles:
\[
45^\circ + 80^\circ + x = 180^\circ
\]

Simplify:
\[
125^\circ + x = 180^\circ
\]

Solve for \( x \):
\[
x = 180^\circ - 125^\circ = 55^\circ
\]

Answer for Problem 8:
\[
\boxed{55}
\]

---

Final Answers

\[
\boxed{55, 20, 105, 160, 110, 68, 135, 55}
\]