Angles in Polygons Worksheets - Math Monks - Free Printable

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

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