Find the missing angles in irregular polygons using the formula for the sum of interior angles.
Worksheet titled "WORKOUT POLYGONS" with 12 irregular polygon diagrams, each showing some interior angles and one missing angle labeled 'x'. The formula for the sum of interior angles is given as (n-2) × 180°. Students are asked to find the missing angle in each polygon. Answers are provided at the bottom in a table format.
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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 finding interior angles of a polygon worksheet.
