[4] Angles in polygons and parallel lines y x Calculate the value of x, giving reasons 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. 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

[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, 360 – 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 - 12 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 + 10 + 50 = 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 + 18 + 2x + 7 + 2x = 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 180 - 38 - 120 = 22 (triangle) Angle BDC = 22 (alternate) Angle DEC = 139 (straight line) Angle x = 19 (triangle) 360 – 90 – 108 - 108 (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 – 108 - 108 (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

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

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)

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)

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

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 – 108 - 108 (around a point) 84

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)

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

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 – 108 - 108 (around a point) 54

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

10 o’clock Find the size of the largest angle in the triangle. x + 18 + 2x + 7 + 2x = 180 or 5x + 25 = 180 5x = 155 or 5x = 180 – 25 or x = (Their 155) ÷ 5 = 31 2 × (Their x) + 7  = 69

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

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