Math worksheet for learning about angles and polygons.

Here are the step-by-step solutions to the problems on the worksheet.

1. Find the missing angles in these polygons

To find a missing interior angle, we first calculate the Sum of Interior Angles using the formula:
$$Sum = (n - 2) \times 180^{\circ}$$
*(where $n$ is the number of sides)*. Then, subtract the known angles from this sum.

a) Hexagon (6 sides)
* Step 1: Calculate total sum. $(6 - 2) \times 180^{\circ} = 4 \times 180^{\circ} = 720^{\circ}$.
* Step 2: Add the known angles: $90 + 140 + 135 + 85 + 160 = 610^{\circ}$.
* Step 3: Subtract from total: $720^{\circ} - 610^{\circ} = 110^{\circ}$.
* Answer: $x = 110^{\circ}$

b) Heptagon (7 sides)
* Step 1: Calculate total sum. $(7 - 2) \times 180^{\circ} = 5 \times 180^{\circ} = 900^{\circ}$.
* Step 2: Add the known angles: $100 + 130 + 115 + 40 + 50 = 435^{\circ}$.
* Step 3: Subtract from total: $900^{\circ} - 435^{\circ} = 465^{\circ}$.
* Answer: $x = 465^{\circ}$ *(Note: This is a reflex angle, which is greater than 180°)*

c) Octagon (8 sides)
* Step 1: Calculate total sum. $(8 - 2) \times 180^{\circ} = 6 \times 180^{\circ} = 1080^{\circ}$.
* Step 2: Add the known angles: $95 + 85 + 275 + 280 + 140 + 91 + 55 = 1021^{\circ}$.
* Step 3: Subtract from total: $1080^{\circ} - 1021^{\circ} = 59^{\circ}$.
* Answer: $x = 59^{\circ}$

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2. Polygon Sums and Sides

a) Sum of angles in a 20-sided polygon
* Use the formula: $(n - 2) \times 180^{\circ}$.
* Calculation: $(20 - 2) \times 180^{\circ} = 18 \times 180^{\circ} = 3240^{\circ}$.
* Answer: $3240^{\circ}$

b) How many sides for an angle sum of 2700°?
* Set up the equation: $(n - 2) \times 180 = 2700$.
* Divide by 180: $n - 2 = 15$.
* Add 2: $n = 17$.
* Answer: 17 sides

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3. Regular Polygons

a) Name each polygon
Count the sides for each shape from left to right:
1. 5 sides: Pentagon
2. 8 sides: Octagon
3. 9 sides: Nonagon
4. 10 sides: Decagon
5. 6 sides: Hexagon

b) Size of ONE interior angle
Formula: $\frac{(n-2) \times 180}{n}$
1. Pentagon: $\frac{540}{5} = \mathbf{108^{\circ}}$
2. Octagon: $\frac{1080}{8} = \mathbf{135^{\circ}}$
3. Nonagon: $\frac{1260}{9} = \mathbf{140^{\circ}}$
4. Decagon: $\frac{1440}{10} = \mathbf{144^{\circ}}$
5. Hexagon: $\frac{720}{6} = \mathbf{120^{\circ}}$

c) Size of ONE exterior angle
Formula: $\frac{360}{n}$
1. Pentagon: $\frac{360}{5} = \mathbf{72^{\circ}}$
2. Octagon: $\frac{360}{8} = \mathbf{45^{\circ}}$
3. Nonagon: $\frac{360}{9} = \mathbf{40^{\circ}}$
4. Decagon: $\frac{360}{10} = \mathbf{36^{\circ}}$
5. Hexagon: $\frac{360}{6} = \mathbf{60^{\circ}}$

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4. Find the number of sides

We use the exterior angle rule ($n = \frac{360}{\text{exterior angle}}$) or the interior angle relationship ($\text{Exterior} = 180 - \text{Interior}$).

a) Exterior angles of 36°
* $n = \frac{360}{36} = 10$
* Answer: 10 sides

b) Exterior angles of 30°
* $n = \frac{360}{30} = 12$
* Answer: 12 sides

c) Exterior angles of 9°
* $n = \frac{360}{9} = 40$
* Answer: 40 sides

d) Interior angles of 156°
* First find exterior: $180 - 156 = 24^{\circ}$.
* Then find sides: $\frac{360}{24} = 15$.
* Answer: 15 sides

e) Interior angles of 168°
* First find exterior: $180 - 168 = 12^{\circ}$.
* Then find sides: $\frac{360}{12} = 30$.
* Answer: 30 sides

f) Interior angles of 168.75°
* First find exterior: $180 - 168.75 = 11.25^{\circ}$.
* Then find sides: $\frac{360}{11.25} = 32$.
* Answer: 32 sides

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5. Find the missing angles (Regular Shapes)

The angle marked is usually the "gap" between two shapes joined at a vertex. The angles around a single point always add up to $360^{\circ}$.

a) Square and Hexagon
* Interior angle of Square: $90^{\circ}$.
* Interior angle of Hexagon: $120^{\circ}$.
* Equation: $90 + 120 + a = 360$.
* $210 + a = 360 \rightarrow a = 150^{\circ}$.
* Answer: $a = 150^{\circ}$

b) Pentagon, Octagon, and Square
* Interior angle of Pentagon: $108^{\circ}$.
* Interior angle of Octagon: $135^{\circ}$.
* Interior angle of Square: $90^{\circ}$.
* Equation: $108 + 135 + 90 + b = 360$.
* $333 + b = 360 \rightarrow b = 27^{\circ}$.
* Answer: $b = 27^{\circ}$

c) Decagon, Square, and Pentagon
* Interior angle of Decagon: $144^{\circ}$.
* Interior angle of Square: $90^{\circ}$.
* Interior angle of Pentagon: $108^{\circ}$.
* Equation: $144 + 90 + 108 + c = 360$.
* $342 + c = 360 \rightarrow c = 18^{\circ}$.
* Answer: $c = 18^{\circ}$

Final Answer:
1. a) 110°, b) 465°, c) 59°
2. a) 3240°, b) 17 sides
3. a) Pentagon, Octagon, Nonagon, Decagon, Hexagon
b) 108°, 135°, 140°, 144°, 120°
c) 72°, 45°, 40°, 36°, 60°
4. a) 10, b) 12, c) 40, d) 15, e) 30, f) 32
5. a) 150°, b) 27°, c) 18°