Interior And Exterior Angle Measures Worksheet - 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:
\[
\text{Sum of Interior Angles} = (n-2) \times 180^\circ
\]

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 Exterior Angles: The sum of the exterior angles of any polygon is always \( 360^\circ \).

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

---

Problem 1

The polygon is a quadrilateral (4 sides). The sum of the interior angles is:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
The 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, the answer is:
\[
\boxed{55}
\]

---

Problem 2

The polygon is a triangle (3 sides). The sum of the interior angles is:
\[
(3-2) \times 180^\circ = 1 \times 180^\circ = 180^\circ
\]
The 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 the sum of the angles in a triangle cannot exceed \( 180^\circ \). Let's recheck the problem: The angle \( x \) is actually the exterior angle at the vertex where the two given angles meet. The exterior angle is supplementary to the interior angle, so:
\[
x = 180^\circ - 100^\circ = 80^\circ
\]
So, the answer is:
\[
\boxed{80}
\]

---

Problem 3

The polygon is a quadrilateral (4 sides). The sum of the interior angles is:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
The 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, the answer is:
\[
\boxed{105}
\]

---

Problem 4

The polygon is a quadrilateral (4 sides). The sum of the interior angles is:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
The given angles are \( 40^\circ \), \( 60^\circ \), \( 70^\circ \), and \( x^\circ \). We can set up the equation:
\[
40^\circ + 60^\circ + 70^\circ + x^\circ = 360^\circ
\]
Simplify:
\[
170^\circ + x^\circ = 360^\circ
\]
Solve for \( x \):
\[
x = 360^\circ - 170^\circ = 190^\circ
\]
So, the answer is:
\[
\boxed{190}
\]

---

Problem 5

The polygon is a quadrilateral (4 sides). The sum of the interior angles is:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
The given angles are \( 55^\circ \), \( 102^\circ \), \( 93^\circ \), and \( x^\circ \). We can set up the equation:
\[
55^\circ + 102^\circ + 93^\circ + x^\circ = 360^\circ
\]
Simplify:
\[
250^\circ + x^\circ = 360^\circ
\]
Solve for \( x \):
\[
x = 360^\circ - 250^\circ = 110^\circ
\]
So, the answer is:
\[
\boxed{110}
\]

---

Problem 6

The polygon is a triangle (3 sides). The sum of the interior angles is:
\[
(3-2) \times 180^\circ = 1 \times 180^\circ = 180^\circ
\]
The 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, the answer is:
\[
\boxed{58}
\]

---

Problem 7

The polygon is a pentagon (5 sides). The sum of the interior angles is:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
The given angles are \( 168^\circ \), \( 150^\circ \), \( 63^\circ \), \( 75^\circ \), and \( x^\circ \). We can set up the equation:
\[
168^\circ + 150^\circ + 63^\circ + 75^\circ + x^\circ = 540^\circ
\]
Simplify:
\[
456^\circ + x^\circ = 540^\circ
\]
Solve for \( x \):
\[
x = 540^\circ - 456^\circ = 84^\circ
\]
So, the answer is:
\[
\boxed{84}
\]

---

Problem 8

The polygon is a triangle (3 sides). The sum of the interior angles is:
\[
(3-2) \times 180^\circ = 1 \times 180^\circ = 180^\circ
\]
The 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, the answer is:
\[
\boxed{55}
\]

---

Final Answers:

\[
\boxed{55, 80, 105, 190, 110, 58, 84, 55}
\]