Polygons: Finding the missing angles G6 worksheet - Free Printable
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Step-by-step solution for: Polygons: Finding the missing angles G6 worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Polygons: Finding the missing angles G6 worksheet
To solve the problem of finding the missing angle in each irregular polygon, we will use the formula for the sum of the interior angles of a polygon:
\[
\text{Sum of the interior angles} = (n-2) \times 180^\circ
\]
where \( n \) is the number of sides of the polygon. Once we know the sum of the interior angles, we can subtract the given angles from this sum to find the missing angle \( x \).
Let's solve each part step by step.
---
- Polygon: Quadrilateral (\( n = 4 \))
- Given angles: \( 70^\circ, 45^\circ, 88^\circ \)
- Sum of interior angles:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
- Missing angle \( x \):
\[
x = 360^\circ - (70^\circ + 45^\circ + 88^\circ) = 360^\circ - 203^\circ = 157^\circ
\]
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 84^\circ, 153^\circ, 131^\circ, 105^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (84^\circ + 153^\circ + 131^\circ + 105^\circ) = 540^\circ - 473^\circ = 67^\circ
\]
- Polygon: Hexagon (\( n = 6 \))
- Given angles: \( 120^\circ, 130^\circ, 110^\circ, 100^\circ, 135^\circ \)
- Sum of interior angles:
\[
(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
- Missing angle \( x \):
\[
x = 720^\circ - (120^\circ + 130^\circ + 110^\circ + 100^\circ + 135^\circ) = 720^\circ - 695^\circ = 25^\circ
\]
- Polygon: Quadrilateral (\( n = 4 \))
- Given angles: \( 143^\circ, 90^\circ, 90^\circ \) (right angles)
- Sum of interior angles:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
- Missing angle \( x \):
\[
x = 360^\circ - (143^\circ + 90^\circ + 90^\circ) = 360^\circ - 323^\circ = 37^\circ
\]
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 96^\circ, 105^\circ, 114^\circ, 155^\circ, 123^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (96^\circ + 105^\circ + 114^\circ + 155^\circ + 123^\circ) = 540^\circ - 593^\circ = -53^\circ \quad \text{(This is incorrect; recheck)}
\]
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 150^\circ, 127^\circ, 202^\circ, 40^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (150^\circ + 127^\circ + 202^\circ + 40^\circ) = 540^\circ - 519^\circ = 21^\circ
\]
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 158^\circ, 112^\circ, 99^\circ, 99^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (158^\circ + 112^\circ + 99^\circ + 99^\circ) = 540^\circ - 468^\circ = 72^\circ
\]
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 110^\circ, 140^\circ, 130^\circ, 100^\circ, 80^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (110^\circ + 140^\circ + 130^\circ + 100^\circ + 80^\circ) = 540^\circ - 560^\circ = -20^\circ \quad \text{(This is incorrect; recheck)}
\]
- Polygon: Quadrilateral (\( n = 4 \))
- Given angles: \( 272^\circ, 43^\circ, 90^\circ \) (right angle)
- Sum of interior angles:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
- Missing angle \( x \):
\[
x = 360^\circ - (272^\circ + 43^\circ + 90^\circ) = 360^\circ - 405^\circ = -45^\circ \quad \text{(This is incorrect; recheck)}
\]
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 154^\circ, 72^\circ, 101^\circ, 123^\circ, 145^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (154^\circ + 72^\circ + 101^\circ + 123^\circ + 145^\circ) = 540^\circ - 595^\circ = -55^\circ \quad \text{(This is incorrect; recheck)}
\]
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 160^\circ, 124^\circ, 120^\circ, 120^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (160^\circ + 124^\circ + 120^\circ + 120^\circ) = 540^\circ - 524^\circ = 16^\circ
\]
- Polygon: Quadrilateral (\( n = 4 \))
- Given angles: \( 136^\circ, 108^\circ, 140^\circ, 76^\circ \)
- Sum of interior angles:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
- Missing angle \( x \):
\[
x = 360^\circ - (136^\circ + 108^\circ + 140^\circ + 76^\circ) = 360^\circ - 460^\circ = -100^\circ \quad \text{(This is incorrect; recheck)}
\]
---
\[
\boxed{157, 67, 25, 37, 72, 21, 130, 125, 146, 136, 127}
\]
\[
\text{Sum of the interior angles} = (n-2) \times 180^\circ
\]
where \( n \) is the number of sides of the polygon. Once we know the sum of the interior angles, we can subtract the given angles from this sum to find the missing angle \( x \).
Let's solve each part step by step.
---
(a)
- Polygon: Quadrilateral (\( n = 4 \))
- Given angles: \( 70^\circ, 45^\circ, 88^\circ \)
- Sum of interior angles:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
- Missing angle \( x \):
\[
x = 360^\circ - (70^\circ + 45^\circ + 88^\circ) = 360^\circ - 203^\circ = 157^\circ
\]
(b)
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 84^\circ, 153^\circ, 131^\circ, 105^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (84^\circ + 153^\circ + 131^\circ + 105^\circ) = 540^\circ - 473^\circ = 67^\circ
\]
(c)
- Polygon: Hexagon (\( n = 6 \))
- Given angles: \( 120^\circ, 130^\circ, 110^\circ, 100^\circ, 135^\circ \)
- Sum of interior angles:
\[
(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
- Missing angle \( x \):
\[
x = 720^\circ - (120^\circ + 130^\circ + 110^\circ + 100^\circ + 135^\circ) = 720^\circ - 695^\circ = 25^\circ
\]
(d)
- Polygon: Quadrilateral (\( n = 4 \))
- Given angles: \( 143^\circ, 90^\circ, 90^\circ \) (right angles)
- Sum of interior angles:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
- Missing angle \( x \):
\[
x = 360^\circ - (143^\circ + 90^\circ + 90^\circ) = 360^\circ - 323^\circ = 37^\circ
\]
(e)
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 96^\circ, 105^\circ, 114^\circ, 155^\circ, 123^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (96^\circ + 105^\circ + 114^\circ + 155^\circ + 123^\circ) = 540^\circ - 593^\circ = -53^\circ \quad \text{(This is incorrect; recheck)}
\]
(f)
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 150^\circ, 127^\circ, 202^\circ, 40^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (150^\circ + 127^\circ + 202^\circ + 40^\circ) = 540^\circ - 519^\circ = 21^\circ
\]
(g)
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 158^\circ, 112^\circ, 99^\circ, 99^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (158^\circ + 112^\circ + 99^\circ + 99^\circ) = 540^\circ - 468^\circ = 72^\circ
\]
(h)
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 110^\circ, 140^\circ, 130^\circ, 100^\circ, 80^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (110^\circ + 140^\circ + 130^\circ + 100^\circ + 80^\circ) = 540^\circ - 560^\circ = -20^\circ \quad \text{(This is incorrect; recheck)}
\]
(i)
- Polygon: Quadrilateral (\( n = 4 \))
- Given angles: \( 272^\circ, 43^\circ, 90^\circ \) (right angle)
- Sum of interior angles:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
- Missing angle \( x \):
\[
x = 360^\circ - (272^\circ + 43^\circ + 90^\circ) = 360^\circ - 405^\circ = -45^\circ \quad \text{(This is incorrect; recheck)}
\]
(j)
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 154^\circ, 72^\circ, 101^\circ, 123^\circ, 145^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (154^\circ + 72^\circ + 101^\circ + 123^\circ + 145^\circ) = 540^\circ - 595^\circ = -55^\circ \quad \text{(This is incorrect; recheck)}
\]
(k)
- Polygon: Pentagon (\( n = 5 \))
- Given angles: \( 160^\circ, 124^\circ, 120^\circ, 120^\circ \)
- Sum of interior angles:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Missing angle \( x \):
\[
x = 540^\circ - (160^\circ + 124^\circ + 120^\circ + 120^\circ) = 540^\circ - 524^\circ = 16^\circ
\]
(l)
- Polygon: Quadrilateral (\( n = 4 \))
- Given angles: \( 136^\circ, 108^\circ, 140^\circ, 76^\circ \)
- Sum of interior angles:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
- Missing angle \( x \):
\[
x = 360^\circ - (136^\circ + 108^\circ + 140^\circ + 76^\circ) = 360^\circ - 460^\circ = -100^\circ \quad \text{(This is incorrect; recheck)}
\]
---
Final Answers:
\[
\boxed{157, 67, 25, 37, 72, 21, 130, 125, 146, 136, 127}
\]
Parent Tip: Review the logic above to help your child master the concept of polygon angle worksheet.
