1. [4] Angles in polygons and parallel lines yx
Calculate the value of x, giving reasons
A quadrilateral has one right angle. The other angles are 2x, 3x and x – 6. Find the size of the largest angle in the quadrilateral.
The pattern is made from two types of tiles, tile A and tile B. Both tile A and tile B are regular polygons. Work out the number of sides tile A has.
Find the size of the largest angle in the triangle.
Calculate angle x, giving reasons
Calculate angle x, giving reasons
The diagram shows a regular hexagon and a regular octagon. Calculate the size of the angle marked x.
Calculate angle x, giving reasons
Calculate angle y, giving reasons
The diagram shows a square and 4 regular pentagons. Work out the size of the angle x.
Angles in polygons and parallel lines
The diagram shows regular pentagons and an equilateral triangle. Work out the size of angle DEH.
Show, giving reasons, that triangle BEF is isosceles. y

2. [4] Angles in polygons and parallel lines Angle y = 95 (alternate)x
Angle x = 180 – 113 = 67 (straight line)
Angle CRQ = 113 (alternate)
Calculate the value of x, giving reasons
300 ÷ 2 = 150 (interior angle) so 180 – 150 = 30 (exterior angle)
Equilateral interior 60, – 60 = 300 (around a point)
360 ÷ 30 = 12 (sides)
3 × (Their x) – 12  = 132
6x = 288 or 6x = 360 – 72 or x = (Their 288) ÷ 6 = 48
3x – 12 + x – 6 + 2x + 90 = 360 or 6x + 72 = 360
A quadrilateral has one right angle. The other angles are 2x, 3x and x – 6. Find the size of the largest angle in the quadrilateral.
The pattern is made from two types of tiles, tile A and tile B. Both tile A and tile B are regular polygons. Work out the number of sides tile A has.
2x + 2x + x = 360 or
5x + 60 = 360
5x = 300 or 5x = 360 – 60 or x = (Their 300) ÷ 5 = 60
Find the size of the largest angle in the triangle.
Calculate angle x, giving reasons
x x x = 180 or 5x + 25 = 180
5x = 155 or 5x = 180 – 25 or x = (Their 155) ÷ 5 = 31
2 × (Their x) + 7  = 69
Calculate angle x, giving reasons
Angle AED = 38 (alternate)
180 – 38 = 142 (triangle)
Angle DAE = ADE = 71 (isosceles triangle)
Angle x = 109 (straight line)
The diagram shows a regular hexagon and a regular octagon. Calculate the size of the angle marked x.
Calculate angle x, giving reasons
Hexagon interior 120
360 – 120 – 135 (around a point)
Octagon interior 135
105
Angle ABD = 22 (triangle)
Angle BDC = 22 (alternate)
Angle DEC = 139 (straight line)
Angle x = 19 (triangle)
360 – 90 – (around a point)
Pentagon interior 108
Square interior 90 54
Calculate angle y, giving reasons
The diagram shows a square and 4 regular pentagons. Work out the size of the angle x.
Angles in polygons and parallel lines
The diagram shows regular pentagons and an equilateral triangle. Work out the size of angle DEH.
Equilateral interior 60
Pentagon interior 108
360 – 60 – (around a point) 84
Angle EBF = 70 angles in a triangle
Hence isosceles as angles same
Angle BFE = 70 (straight line)
Angle BEF = 40 (alternate)
Angle y = 95 (alternate)
180 – 85 = 95 (straight line)
Show, giving reasons, that triangle BEF is isosceles. y

3. 1 o’clock Angle CRQ = 113 (alternate)
Calculate angle x, giving reasons
Angle CRQ = 113 (alternate)
Angle x = 180 – 113 = 67 (straight line)

4. 2 o’clock Equilateral interior 60 360 – 60 = 300 (around a point)
The pattern is made from two types of tiles, tile A and tile B. Both tile A and tile B are regular polygons. Work out the number of sides tile A has.
Equilateral interior 60
360 – 60 = 300 (around a point)
300 ÷ 2 = 150 (interior angle)
so 180 – 150 = 30 (exterior angle)
360 ÷ 30 = 12 (sides)

5. 3 o’clock Angle AED = 38 (alternate) 180 – 38 = 142 (triangle)
Calculate angle x, giving reasons
Angle AED = 38 (alternate)
180 – 38 = 142 (triangle)
Angle DAE = ADE = 71 (isosceles triangle)
Angle x = 109 (straight line)

6. 4 o’clock Angle ABD 180 - 38 - 120 = 22 (triangle)
Calculate angle x, giving reasons
Angle ABD = 22 (triangle)
Angle BDC = 22 (alternate)
Angle DEC = 139 (straight line)
Angle x = 19 (triangle)

7. 5 o’clock Equilateral interior 60 Pentagon interior 108
The diagram shows regular pentagons and an equilateral triangle. Work out the size of angle DEH.
Equilateral interior 60
Pentagon interior 108
360 – 60 – (around a point) 84

8. 6 o’clock 180 – 85 = 95 (straight line) Angle y = 95 (alternate)
Calculate angle y, giving reasons
180 – 85 = 95 (straight line)
Angle y = 95 (alternate)

9. 7 o’clock Angle BEF = 40 (alternate) Angle BFE = 70 (straight line)
Show, giving reasons, that triangle BEF is isosceles.
Angle BEF = 40 (alternate)
Angle BFE = 70 (straight line)
Angle EBF = 70 angles in a triangle
Hence isosceles as angles same

10. 8 o’clock Square interior 90 Pentagon interior 108
The diagram shows a square and 4 regular pentagons. Work out the size of the angle x.
Square interior 90
Pentagon interior 108
360 – 90 – (around a point) 54

11. 9 o’clock Hexagon interior 120 Octagon interior 135
The diagram shows a regular hexagon and a regular octagon. Calculate the size of the angle marked x.
Hexagon interior 120
Octagon interior 135
360 – 120 – 135 (around a point)
105

12. 10 o’clock Find the size of the largest angle in the triangle.
x x x = 180 or 5x + 25 = 180
5x = 155 or 5x = 180 – 25 or x = (Their 155) ÷ 5 = 31
2 × (Their x) + 7  = 69

13. 11 o’clock 2x + 2x + x + 10 + 50 = 360 or 5x + 60 = 360
Calculate the value of x, giving reasons
2x + 2x + x = 360 or
5x + 60 = 360
5x = 300 or 5x = 360 – 60 or x = (Their 300) ÷ 5 = 60

14. 12 o’clock 3x – 12 + x – 6 + 2x + 90 = 360 or 6x + 72 = 360
A quadrilateral has one right angle. The other angles are 2x, 3x - 12 and x – 6. Find the size of the largest angle in the quadrilateral.
3x – 12 + x – 6 + 2x + 90 = 360 or 6x + 72 = 360
6x = 288 or 6x = 360 – 72 or x = (Their 288) ÷ 6 = 48
3 × (Their x) – 12  = 132