This geometry worksheet helps students calculate interior and exterior angles for various regular polygons, from pentagons to decagons.

Here are the step-by-step solutions for the worksheet.

Section A

1) Find the size of each interior angle in a regular hexagon.
* Step 1: A hexagon has 6 sides ($n=6$).
* Step 2: The sum of interior angles is $(n - 2) \times 180^{\circ}$. So, $(6 - 2) \times 180 = 4 \times 180 = 720^{\circ}$.
* Step 3: Divide by the number of sides to find one angle: $720 / 6 = 120^{\circ}$.
* Answer: $120^{\circ}$

2) Find the size of each interior angle in a regular nonagon.
* Step 1: A nonagon has 9 sides ($n=9$).
* Step 2: Sum of angles = $(9 - 2) \times 180 = 7 \times 180 = 1260^{\circ}$.
* Step 3: One angle = $1260 / 9 = 140^{\circ}$.
* Answer: $140^{\circ}$

3) Find the size of each exterior angle in a regular dodecagon.
* Step 1: A dodecagon has 12 sides ($n=12$).
* Step 2: The sum of exterior angles is always $360^{\circ}$.
* Step 3: Divide by the number of sides: $360 / 12 = 30^{\circ}$.
* Answer: $30^{\circ}$

4) Write an expression for the sum of the interior angles in a polygon with n sides.
* Rule: Subtract 2 from the number of sides and multiply by 180.
* Answer: $(n - 2) \times 180^{\circ}$

5) Write an expression for the size of each interior angle in a polygon with n sides.
* Rule: Take the total sum from question 4 and divide by $n$.
* Answer: $\frac{(n - 2) \times 180}{n}$

6) Write an expression for the size of each exterior angle in a polygon with n sides.
* Rule: Divide 360 by the number of sides.
* Answer: $\frac{360}{n}$

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Section B

1) Regular Pentagon (5 sides)
* Angle a (Interior):
* Interior angle = $(5-2) \times 180 / 5 = 540 / 5 = 108^{\circ}$.
* a = $108^{\circ}$
* Angle b (Exterior):
* Exterior angle = $360 / 5 = 72^{\circ}$. (Or $180 - 108 = 72$).
* b = $72^{\circ}$

2) Regular Pentagon split into triangles
* Angle c: This is part of the top triangle formed by drawing a diagonal. The top vertex angle is the full interior angle ($108^{\circ}$). The triangle is isosceles (two equal sides). The base angles are equal.
* Calculation: $(180 - 108) / 2 = 36^{\circ}$.
* c = $36^{\circ}$
* Angle d: This is the other part of the corner angle. Since the whole corner is $108^{\circ}$ and angle $c$ is $36^{\circ}$:
* Calculation: $108 - 36 = 72^{\circ}$.
* d = $72^{\circ}$

3) Regular Hexagon (6 sides)
* Angle e: This is the full interior angle of a hexagon.
* Calculation: $(6-2) \times 180 / 6 = 120^{\circ}$.
* e = $120^{\circ}$
* Angle f: This angle is inside a triangle formed by two diagonals meeting at a vertex. The triangle is isosceles with the top angle being $120^{\circ}$.
* Calculation: $(180 - 120) / 2 = 30^{\circ}$.
* f = $30^{\circ}$

4) Regular Hexagon divided into equilateral triangles
* Angle g (Exterior): Exterior angle of a hexagon.
* Calculation: $360 / 6 = 60^{\circ}$.
* g = $60^{\circ}$
* Angle h: This is the angle where two equilateral triangles meet at the center. An equilateral triangle has $60^{\circ}$ angles. Angle $h$ combines two of them.
* Calculation: $60 + 60 = 120^{\circ}$.
* h = $120^{\circ}$

5) Regular Octagon (8 sides)
* Angle i: This is the interior angle of a regular octagon.
* Calculation: $(8-2) \times 180 / 8 = 135^{\circ}$.
* i = $135^{\circ}$
* Angle j: This angle is part of a triangle formed by diagonals. The "pointy" triangle at the bottom has a tip angle of $135^{\circ}$. It is isosceles.
* Base angles calculation: $(180 - 135) / 2 = 22.5^{\circ}$. Angle $j$ is one of these base angles.
* j = $22.5^{\circ}$

6) Regular Nonagon (9 sides)
* Angle k: This is the interior angle of a regular nonagon.
* Calculation: $(9-2) \times 180 / 9 = 140^{\circ}$.
* k = $140^{\circ}$
* Angle l: This angle is formed by diagonals connecting vertices 1-4 and 2-5 (skipping 2 vertices). This creates an isosceles triangle with the tip at the top vertex. The tip angle is the interior angle ($140^{\circ}$).
* Base angles calculation: $(180 - 140) / 2 = 20^{\circ}$.
* Angle $l$ is made of two of these base angles meeting together.
* Calculation: $20 + 20 = 40^{\circ}$.
* l = $40^{\circ}$

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

Section B
1) a = $108^{\circ}$, b = $72^{\circ}$
2) c = $36^{\circ}$, d = $72^{\circ}$
3) e = $120^{\circ}$, f = $30^{\circ}$
4) g = $60^{\circ}$, h = $120^{\circ}$
5) i = $135^{\circ}$, j = $22.5^{\circ}$
6) k = $140^{\circ}$, l = $40^{\circ}$