Math worksheet for finding interior and exterior angles in regular polygons, featuring problems and diagrams.

Problem Analysis and Solution

The worksheet focuses on finding angles in regular polygons. Let's solve each section step by step.

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#### Section A: Finding Angles in Regular Polygons

1. Find the size of each interior angle in a regular hexagon.

- Formula for the interior angle of a regular polygon:
\[
\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}
\]
where \( n \) is the number of sides.
- For a hexagon (\( n = 6 \)):
\[
\text{Interior angle} = \frac{(6-2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = \frac{720^\circ}{6} = 120^\circ
\]

Answer: \( 120^\circ \)

2. Find the size of each interior angle in a regular nonagon.

- For a nonagon (\( n = 9 \)):
\[
\text{Interior angle} = \frac{(9-2) \times 180^\circ}{9} = \frac{7 \times 180^\circ}{9} = \frac{1260^\circ}{9} = 140^\circ
\]

Answer: \( 140^\circ \)

3. Find the size of each exterior angle in a regular dodecagon.

- Formula for the exterior angle of a regular polygon:
\[
\text{Exterior angle} = \frac{360^\circ}{n}
\]
- For a dodecagon (\( n = 12 \)):
\[
\text{Exterior angle} = \frac{360^\circ}{12} = 30^\circ
\]

Answer: \( 30^\circ \)

4. Write an expression for the sum of the interior angles in a polygon with \( n \) sides.

- Formula for the sum of the interior angles:
\[
\text{Sum of interior angles} = (n-2) \times 180^\circ
\]

Answer: \( (n-2) \times 180^\circ \)

5. Write an expression for the size of each interior angle in a polygon with \( n \) sides.

- Formula for the interior angle:
\[
\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}
\]

Answer: \( \frac{(n-2) \times 180^\circ}{n} \)

6. Write an expression for the size of each exterior angle in a polygon with \( n \) sides.

- Formula for the exterior angle:
\[
\text{Exterior angle} = \frac{360^\circ}{n}
\]

Answer: \( \frac{360^\circ}{n} \)

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#### Section B: Find the angles indicated. Each question contains one or more regular polygons.

1. Figure 1:
- The figure shows a regular pentagon.
- Interior angle of a pentagon (\( n = 5 \)):
\[
\text{Interior angle} = \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ
\]
- Exterior angle:
\[
\text{Exterior angle} = 180^\circ - 108^\circ = 72^\circ
\]
- Therefore:
\[
a = 108^\circ, \quad b = 72^\circ
\]

Answers: \( a = 108^\circ \), \( b = 72^\circ \)

2. Figure 2:
- The figure shows a regular pentagon.
- Interior angle of a pentagon (\( n = 5 \)):
\[
c = 108^\circ
\]
- The angle \( d \) is an exterior angle:
\[
d = 72^\circ
\]

Answers: \( c = 108^\circ \), \( d = 72^\circ \)

3. Figure 3:
- The figure shows a regular hexagon.
- Interior angle of a hexagon (\( n = 6 \)):
\[
e = 120^\circ
\]
- The angle \( f \) is an exterior angle:
\[
f = 60^\circ
\]

Answers: \( e = 120^\circ \), \( f = 60^\circ \)

4. Figure 4:
- The figure shows a regular octagon.
- Interior angle of an octagon (\( n = 8 \)):
\[
\text{Interior angle} = \frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ
\]
- The angle \( g \) is half of the interior angle (since it is formed by a diagonal):
\[
g = \frac{135^\circ}{2} = 67.5^\circ
\]
- The angle \( h \) is the supplement of \( g \):
\[
h = 180^\circ - 67.5^\circ = 112.5^\circ
\]

Answers: \( g = 67.5^\circ \), \( h = 112.5^\circ \)

5. Figure 5:
- The figure shows a regular hexagon.
- Interior angle of a hexagon (\( n = 6 \)):
\[
\text{Interior angle} = 120^\circ
\]
- The angle \( i \) is half of the interior angle:
\[
i = \frac{120^\circ}{2} = 60^\circ
\]
- The angle \( j \) is the supplement of \( i \):
\[
j = 180^\circ - 60^\circ = 120^\circ
\]

Answers: \( i = 60^\circ \), \( j = 120^\circ \)

6. Figure 6:
- The figure shows a regular octagon.
- Interior angle of an octagon (\( n = 8 \)):
\[
\text{Interior angle} = 135^\circ
\]
- The angle \( k \) is half of the interior angle:
\[
k = \frac{135^\circ}{2} = 67.5^\circ
\]
- The angle \( l \) is the supplement of \( k \):
\[
l = 180^\circ - 67.5^\circ = 112.5^\circ
\]

Answers: \( k = 67.5^\circ \), \( l = 112.5^\circ \)

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Final Answers:

Section A:
1. \( 120^\circ \)
2. \( 140^\circ \)
3. \( 30^\circ \)
4. \( (n-2) \times 180^\circ \)
5. \( \frac{(n-2) \times 180^\circ}{n} \)
6. \( \frac{360^\circ}{n} \)

Section B:
1. \( a = 108^\circ \), \( b = 72^\circ \)
2. \( c = 108^\circ \), \( d = 72^\circ \)
3. \( e = 120^\circ \), \( f = 60^\circ \)
4. \( g = 67.5^\circ \), \( h = 112.5^\circ \)
5. \( i = 60^\circ \), \( j = 120^\circ \)
6. \( k = 67.5^\circ \), \( l = 112.5^\circ \)

\boxed{
\begin{array}{l}
\text{Section A:} \\
1. 120^\circ \\
2. 140^\circ \\
3. 30^\circ \\
4. (n-2) \times 180^\circ \\
5. \frac{(n-2) \times 180^\circ}{n} \\
6. \frac{360^\circ}{n} \\
\text{Section B:} \\
1. a = 108^\circ, b = 72^\circ \\
2. c = 108^\circ, d = 72^\circ \\
3. e = 120^\circ, f = 60^\circ \\
4. g = 67.5^\circ, h = 112.5^\circ \\
5. i = 60^\circ, j = 120^\circ \\
6. k = 67.5^\circ, l = 112.5^\circ \\
\end{array}
}