Angles in Polygons Textbook Exercise - Corbettmaths - Free Printable
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Step-by-step solution for: Angles in Polygons Textbook Exercise - Corbettmaths
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Show Answer Key & Explanations
Step-by-step solution for: Angles in Polygons Textbook Exercise - Corbettmaths
To solve the problem of finding the missing angle in each irregular polygon, we need to use the formula for the sum of the interior angles of a polygon. The formula is:
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]
where \( n \) is the number of sides of the polygon.
Let's solve each part step by step.
---
The polygon is a quadrilateral (\( n = 4 \)).
1. Sum of interior angles:
\[
(4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
2. Given angles: \( 70^\circ, 45^\circ, 88^\circ \)
3. Find the missing angle \( x \):
\[
x + 70^\circ + 45^\circ + 88^\circ = 360^\circ
\]
\[
x + 203^\circ = 360^\circ
\]
\[
x = 360^\circ - 203^\circ = 157^\circ
\]
Answer for (a): \( x = 157^\circ \)
---
The polygon is a pentagon (\( n = 5 \)).
1. Sum of interior angles:
\[
(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
2. Given angles: \( 84^\circ, 153^\circ, 131^\circ, 105^\circ \)
3. Find the missing angle \( x \):
\[
x + 84^\circ + 153^\circ + 131^\circ + 105^\circ = 540^\circ
\]
\[
x + 473^\circ = 540^\circ
\]
\[
x = 540^\circ - 473^\circ = 67^\circ
\]
Answer for (b): \( x = 67^\circ \)
---
The polygon is a hexagon (\( n = 6 \)).
1. Sum of interior angles:
\[
(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
2. Given angles: \( 120^\circ, 130^\circ, 100^\circ, 110^\circ, 135^\circ \)
3. Find the missing angle \( x \):
\[
x + 120^\circ + 130^\circ + 100^\circ + 110^\circ + 135^\circ = 720^\circ
\]
\[
x + 695^\circ = 720^\circ
\]
\[
x = 720^\circ - 695^\circ = 25^\circ
\]
Answer for (c): \( x = 25^\circ \)
---
The polygon is a quadrilateral (\( n = 4 \)) with two right angles (\( 90^\circ \)).
1. Sum of interior angles:
\[
(4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
2. Given angles: \( 90^\circ, 90^\circ, 143^\circ \)
3. Find the missing angle \( x \):
\[
x + 90^\circ + 90^\circ + 143^\circ = 360^\circ
\]
\[
x + 323^\circ = 360^\circ
\]
\[
x = 360^\circ - 323^\circ = 37^\circ
\]
Answer for (d): \( x = 37^\circ \)
---
The polygon is a hexagon (\( n = 6 \)).
1. Sum of interior angles:
\[
(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
2. Given angles: \( 96^\circ, 105^\circ, 114^\circ, 155^\circ, 123^\circ \)
3. Find the missing angle \( x \):
\[
x + 96^\circ + 105^\circ + 114^\circ + 155^\circ + 123^\circ = 720^\circ
\]
\[
x + 693^\circ = 720^\circ
\]
\[
x = 720^\circ - 693^\circ = 27^\circ
\]
Answer for (e): \( x = 27^\circ \)
---
The polygon is a pentagon (\( n = 5 \)).
1. Sum of interior angles:
\[
(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
2. Given angles: \( 150^\circ, 127^\circ, 202^\circ, 40^\circ \)
3. Find the missing angle \( x \):
\[
x + 150^\circ + 127^\circ + 202^\circ + 40^\circ = 540^\circ
\]
\[
x + 519^\circ = 540^\circ
\]
\[
x = 540^\circ - 519^\circ = 21^\circ
\]
Answer for (f): \( x = 21^\circ \)
---
\[
\boxed{157^\circ, 67^\circ, 25^\circ, 37^\circ, 27^\circ, 21^\circ}
\]
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]
where \( n \) is the number of sides of the polygon.
Let's solve each part step by step.
---
(a)
The polygon is a quadrilateral (\( n = 4 \)).
1. Sum of interior angles:
\[
(4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
2. Given angles: \( 70^\circ, 45^\circ, 88^\circ \)
3. Find the missing angle \( x \):
\[
x + 70^\circ + 45^\circ + 88^\circ = 360^\circ
\]
\[
x + 203^\circ = 360^\circ
\]
\[
x = 360^\circ - 203^\circ = 157^\circ
\]
Answer for (a): \( x = 157^\circ \)
---
(b)
The polygon is a pentagon (\( n = 5 \)).
1. Sum of interior angles:
\[
(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
2. Given angles: \( 84^\circ, 153^\circ, 131^\circ, 105^\circ \)
3. Find the missing angle \( x \):
\[
x + 84^\circ + 153^\circ + 131^\circ + 105^\circ = 540^\circ
\]
\[
x + 473^\circ = 540^\circ
\]
\[
x = 540^\circ - 473^\circ = 67^\circ
\]
Answer for (b): \( x = 67^\circ \)
---
(c)
The polygon is a hexagon (\( n = 6 \)).
1. Sum of interior angles:
\[
(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
2. Given angles: \( 120^\circ, 130^\circ, 100^\circ, 110^\circ, 135^\circ \)
3. Find the missing angle \( x \):
\[
x + 120^\circ + 130^\circ + 100^\circ + 110^\circ + 135^\circ = 720^\circ
\]
\[
x + 695^\circ = 720^\circ
\]
\[
x = 720^\circ - 695^\circ = 25^\circ
\]
Answer for (c): \( x = 25^\circ \)
---
(d)
The polygon is a quadrilateral (\( n = 4 \)) with two right angles (\( 90^\circ \)).
1. Sum of interior angles:
\[
(4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
2. Given angles: \( 90^\circ, 90^\circ, 143^\circ \)
3. Find the missing angle \( x \):
\[
x + 90^\circ + 90^\circ + 143^\circ = 360^\circ
\]
\[
x + 323^\circ = 360^\circ
\]
\[
x = 360^\circ - 323^\circ = 37^\circ
\]
Answer for (d): \( x = 37^\circ \)
---
(e)
The polygon is a hexagon (\( n = 6 \)).
1. Sum of interior angles:
\[
(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
2. Given angles: \( 96^\circ, 105^\circ, 114^\circ, 155^\circ, 123^\circ \)
3. Find the missing angle \( x \):
\[
x + 96^\circ + 105^\circ + 114^\circ + 155^\circ + 123^\circ = 720^\circ
\]
\[
x + 693^\circ = 720^\circ
\]
\[
x = 720^\circ - 693^\circ = 27^\circ
\]
Answer for (e): \( x = 27^\circ \)
---
(f)
The polygon is a pentagon (\( n = 5 \)).
1. Sum of interior angles:
\[
(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
2. Given angles: \( 150^\circ, 127^\circ, 202^\circ, 40^\circ \)
3. Find the missing angle \( x \):
\[
x + 150^\circ + 127^\circ + 202^\circ + 40^\circ = 540^\circ
\]
\[
x + 519^\circ = 540^\circ
\]
\[
x = 540^\circ - 519^\circ = 21^\circ
\]
Answer for (f): \( x = 21^\circ \)
---
Final Answers
\[
\boxed{157^\circ, 67^\circ, 25^\circ, 37^\circ, 27^\circ, 21^\circ}
\]
Parent Tip: Review the logic above to help your child master the concept of finding missing angles polygons worksheet.
