Geometry worksheet on polygons with exercises to identify shapes, calculate interior angles, and solve for variables.

Let’s solve each problem step by step.

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Problem 1: Triangle (3 sides)
- Name: Triangle or Equilateral triangle (since it's regular)
- Sum of interior angles = (n - 2) × 180° = (3 - 2) × 180° = 180°
- One interior angle = 180° ÷ 3 = 60°

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Problem 2: Octagon (8 sides)
- Name: Octagon
- Sum of interior angles = (8 - 2) × 180° = 6 × 180° = 1080°
- One interior angle = 1080° ÷ 8 = 135°

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Problem 3: Hexagon (6 sides)
- Name: Hexagon
- Sum of interior angles = (6 - 2) × 180° = 4 × 180° = 720°
- One interior angle = 720° ÷ 6 = 120°

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Problem 4: Pentagon (5 sides)
- Name: Pentagon
- Sum of interior angles = (5 - 2) × 180° = 3 × 180° = 540°
- One interior angle = 540° ÷ 5 = 108°

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Problem 5: Nonagon (9 sides)
- Name: Nonagon (already given)
- Sum of interior angles = (9 - 2) × 180° = 7 × 180° = 1260°
- One interior angle = 1260° ÷ 9 = 140°

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Problem 6: Dodecagon (12 sides)
- Name: Dodecagon (already given)
- Sum of interior angles = (12 - 2) × 180° = 10 × 180° = 1800°
- One interior angle = 1800° ÷ 12 = 150°

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Problem 7: Sum of interior angles = 900°, find number of sides
Use formula: Sum = (n - 2) × 180°
So, 900 = (n - 2) × 180
Divide both sides by 180: 900 ÷ 180 = n - 2 → 5 = n - 2
Add 2: n = 7

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Problem 8: One interior angle = 144°, find number of sides
For a regular polygon, one interior angle = [(n - 2) × 180] / n
Set equal to 144:
[(n - 2) × 180] / n = 144
Multiply both sides by n: (n - 2) × 180 = 144n
Expand: 180n - 360 = 144n
Subtract 144n: 36n - 360 = 0
Add 360: 36n = 360
Divide: n = 10

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Problem 9: One interior angle = 160°, find number of sides
Same method:
[(n - 2) × 180] / n = 160
Multiply by n: (n - 2) × 180 = 160n
Expand: 180n - 360 = 160n
Subtract 160n: 20n - 360 = 0
Add 360: 20n = 360
Divide: n = 18

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Problem 10: Quadrilateral with angles 120°, y°, 85°, 53°
Sum of interior angles in quadrilateral = 360°
So: 120 + y + 85 + 53 = 360
Add knowns: 120 + 85 + 53 = 258
Then: y = 360 - 258 = 102

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Problem 11: Pentagon with angles 105°, 117°, 100°, x°, 115°
Sum of interior angles in pentagon = (5 - 2) × 180 = 540°
Add knowns: 105 + 117 + 100 + 115 = 437
Then: x = 540 - 437 = 103

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Problem 12: Heptagon (7 sides) with angles 129°, 116°, 120°, x°, 125°, 135°, 130°
Sum of interior angles = (7 - 2) × 180 = 5 × 180 = 900°
Add knowns: 129 + 116 + 120 + 125 + 135 + 130 = let’s compute step by step:
129 + 116 = 245
245 + 120 = 365
365 + 125 = 490
490 + 135 = 625
625 + 130 = 755
Then: x = 900 - 755 = 145

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Final Answer:

1. Name: Triangle; Sum: 180°; One angle: 60°
2. Name: Octagon; Sum: 1080°; One angle: 135°
3. Name: Hexagon; Sum: 720°; One angle: 120°
4. Name: Pentagon; Sum: 540°; One angle: 108°
5. Sum: 1260°; One angle: 140°
6. Sum: 1800°; One angle: 150°
7. Number of sides: 7
8. Number of sides: 10
9. Number of sides: 18
10. y = 102
11. x = 103
12. x = 145