Interior And Exterior Angles Worksheet - Free Printable

To solve the problem of finding the unknown angles in each polygon, we need to use the properties of polygons, specifically the sum of interior angles and the relationships between angles. Let's go through each polygon step by step.

1. First Polygon (Quadrilateral)

The given angles are: \(33^\circ\), \(140^\circ\), \(x + 75^\circ\), and \(2x\).

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

Set up the equation:
\[
33 + 140 + (x + 75) + 2x = 360
\]

Simplify:
\[
33 + 140 + x + 75 + 2x = 360
\]
\[
348 + 3x = 360
\]

Solve for \(x\):
\[
3x = 360 - 348
\]
\[
3x = 12
\]
\[
x = 4
\]

So, the unknown angles are:
\[
x + 75 = 4 + 75 = 79^\circ
\]
\[
2x = 2 \times 4 = 8^\circ
\]

2. Second Polygon (Quadrilateral)

The given angles are: \(110^\circ\), \(80^\circ\), \(x\), and \(2x\).

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

Set up the equation:
\[
110 + 80 + x + 2x = 360
\]

Simplify:
\[
190 + 3x = 360
\]

Solve for \(x\):
\[
3x = 360 - 190
\]
\[
3x = 170
\]
\[
x = \frac{170}{3} \approx 56.67^\circ
\]

So, the unknown angles are:
\[
x = 56.67^\circ
\]
\[
2x = 2 \times 56.67 \approx 113.33^\circ
\]

3. Third Polygon (Pentagon)

The given angles are: \(113^\circ\), \(3x + 8^\circ\), \(2x + 9^\circ\), \(4x\), and \(x\).

The sum of the interior angles of a pentagon is:
\[
(5-2) \times 180 = 540^\circ
\]

Set up the equation:
\[
113 + (3x + 8) + (2x + 9) + 4x + x = 540
\]

Simplify:
\[
113 + 3x + 8 + 2x + 9 + 4x + x = 540
\]
\[
113 + 10x + 17 = 540
\]
\[
130 + 10x = 540
\]

Solve for \(x\):
\[
10x = 540 - 130
\]
\[
10x = 410
\]
\[
x = 41
\]

So, the unknown angles are:
\[
3x + 8 = 3 \times 41 + 8 = 123 + 8 = 131^\circ
\]
\[
2x + 9 = 2 \times 41 + 9 = 82 + 9 = 91^\circ
\]
\[
4x = 4 \times 41 = 164^\circ
\]
\[
x = 41^\circ
\]

4. Fourth Polygon (Quadrilateral)

The given angles are: \(64^\circ\), \(102^\circ\), \(122^\circ\), and \(x\).

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

Set up the equation:
\[
64 + 102 + 122 + x = 360
\]

Simplify:
\[
364 + x = 360
\]

Solve for \(x\):
\[
x = 360 - 364
\]
\[
x = -4
\]

This result is not possible for an angle in a polygon. There might be a mistake in the problem setup or interpretation. Let's assume it was meant to be a different configuration or recheck the problem.

5. Fifth Polygon (Quadrilateral)

The given angles are: \(66^\circ\), \(117^\circ\), and \(2y\).

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

Set up the equation:
\[
66 + 117 + 2y + 90 = 360
\]

Simplify:
\[
273 + 2y = 360
\]

Solve for \(y\):
\[
2y = 360 - 273
\]
\[
2y = 87
\]
\[
y = 43.5
\]

So, the unknown angle is:
\[
2y = 2 \times 43.5 = 87^\circ
\]

6. Sixth Polygon (Hexagon)

The given angles are: \(93^\circ\), \(47^\circ\), \(62^\circ\), \(3x\), \(4x\), and \(x\).

The sum of the interior angles of a hexagon is:
\[
(6-2) \times 180 = 720^\circ
\]

Set up the equation:
\[
93 + 47 + 62 + 3x + 4x + x = 720
\]

Simplify:
\[
202 + 8x = 720
\]

Solve for \(x\):
\[
8x = 720 - 202
\]
\[
8x = 518
\]
\[
x = 64.75
\]

So, the unknown angles are:
\[
3x = 3 \times 64.75 = 194.25^\circ
\]
\[
4x = 4 \times 64.75 = 259^\circ
\]
\[
x = 64.75^\circ
\]

7. Seventh Polygon (Triangle)

The given angles are: \(70^\circ\), \(60^\circ\), and \(x\).

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

Set up the equation:
\[
70 + 60 + x = 180
\]

Simplify:
\[
130 + x = 180
\]

Solve for \(x\):
\[
x = 180 - 130
\]
\[
x = 50
\]

8. Eighth Polygon (Quadrilateral)

The given angles are: \(85^\circ\), \(75^\circ\), \(108^\circ\), and \(x\).

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

Set up the equation:
\[
85 + 75 + 108 + x = 360
\]

Simplify:
\[
268 + x = 360
\]

Solve for \(x\):
\[
x = 360 - 268
\]
\[
x = 92
\]

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \quad x = 4, \quad x + 75 = 79^\circ, \quad 2x = 8^\circ \\
2. & \quad x \approx 56.67^\circ, \quad 2x \approx 113.33^\circ \\
3. & \quad x = 41, \quad 3x + 8 = 131^\circ, \quad 2x + 9 = 91^\circ, \quad 4x = 164^\circ \\
4. & \quad \text{Error in problem setup} \\
5. & \quad 2y = 87^\circ \\
6. & \quad x = 64.75, \quad 3x = 194.25^\circ, \quad 4x = 259^\circ \\
7. & \quad x = 50 \\
8. & \quad x = 92 \\
\end{aligned}
}
\]