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EVERYTHING YOU NEED TO KNOW TO GET A GRADE C GEOMETRY & MEASURES (FOUNDATION) Part 1 Rhombus Trapezium Rectangle Rhombus Rhombus Parallelogram Rhombus Trapezium or Rightangle Trapezium 110° 250° Base angles in a kite are equal Opposite angles in a rhombus are equal Angles around a point sum to 360° Angles in a kite sum to 360° 30 Kite Trapezium Replace a with 3 and b with 5.2 P = 2 x 3 + 2 x 5.2 P = 6 + 10.4 Total areas of both shapes are equal to one as shown. 16.4 Equilateral triangle Rhombus 2 (Fits on top of itself twice through a full turn) 5cm 5cm 3cm 3cm 9cm Can choose either as your answer 5cm Any two of rectangle, parallelogram, kite or arrowhead The 3cm and 5cm rods would not meet when joined with the 9cm rod. In an isosceles triangle, the base angles are the same Angles in a triangle sum to 180° 80 50 20 50 Each angle is 60° in an equilateral triangle 120° because angles on a straight line add to 180° 30° because angles in a right angle add to 90° Base angles are both 30° so ABD is an isosceles triangle Isosceles Triangle 35° 73° 107° 73° 107° because angles on a straight line add up to 180° 180° - 34° = 146° 146° ÷ 2 = 73° 73 y = 180° - 107° - 38° = 35° 35 No, because 38° is not equal to 35°. Therefore, it is not an isosceles triangle Angles in a triangle add to 180° Angles in a triangle add to 180° 153° 27° 27° 180° - 126° = 54° 54° ÷ 2 = 27° 153° because angles on a straight line add up to 180° 27 153 Base angles are the same in an isosceles triangle 80° Means work out angle A in triangle ABC Angles in a triangle add to 180° 180° - 80° - 80° = 20° 20 65° 50° Base angles are the same in an isosceles triangle 65° 70° 40° Means work out angle R in triangle PQR Angles in a triangle add to 180° 180° - 70° - 70° = 40° A right angle is 90° 90° - 40° = 50° Both base angles are equal 180° - 50° = 130° 130° ÷ 2 = 65° 65 66° 66° 180° - 48° = 132° 132° ÷ 2 = 66° Angles on a straight line add to 180° 180° - 66° = 114° 114° 114 A quadrilateral is made up of two triangles 180° 180° Angles is a triangle add up to 180° 180° + 180° = 360° LEARN OFF BY HEART (because the exterior angles add up to 360°) Exterior angle = 72° Two exterior angles joined together 72° As worked out in part (a) 72° 144 All the angles and sides are the same in a regular pentagon Exterior 72° angle Exterior 72° angle 36° LEARN OFF BY HEART Interior 108° angle Exterior 72° angle 36° Exterior 72° angle Exterior 72° angle = 72° Angles on a straight line add up 180° Interior angle = 180° - 72° = 108° Base angles in a isosceles triangle are the same 36 180° - 108° = 72° Interior angle LEARN OFF BY HEART Exterior angle Angles on a straight line add up 180° Exterior angle = 180° - 162° = 18° = 20 20 Decagon LEARN OFF BY HEART Pentagon Interior 108° angle Interior 144° angle Sum of interior angles = (number of sides – 2) x 180° 144° Sum of interior angles of an decagon = (10 – 2) x 180° = 8 x 180° = 1440° = 144° Sum of interior angles of a pentagon = (5 – 2) x 180°= 3 x 180° = 540° = 108° Angles around a point add up to 360° 360° - 144° - 108° = 108° Base angles in a isosceles triangle are the same 180° - 108° = 72° 72° ÷ 2 = 36° 144° + 36° = 180° (Angles on a straight line add up to 180°) Therefore, ABC lie on a straight line LEARN OFF BY HEART Sum of interior angles = (number of sides – 2) x 180° Hexagon Interior angle 120° Square 60° Square 60° 60° Sum of interior angles of an hexagon = (6 – 2) x 180° = 4 x 180° = 720° = 120° Angles around a point add up to 360° 360° - 120° - 90° - 90°= 60° Base angles are the same 180° - 60° = 120° 120° ÷ 2= 60° Therefore, as all angles are 60° AHJ is equilateral Sum of interior angles of an octagon = (8 – 2) x 180° = 6 x 180° = 1080° = 135° LEARN OFF BY HEART Sum of interior angles = (number of sides – 2) x 180° 135 135° 135° 135° 135° 135° 135° 135° 135° = 135° Angles around a point add up to 360° 360° - 135° - 135° = 90° Therefore, as all angles are 90° PQRS is a square As worked out in part (a) 2.5 -1 70° 40° 70° Alternate angles are equal Angles on a straight line add up 180° 180° - 110° = 70° Angles in a triangle add up 180° 180° - 40° - 70° = 70° As both base angles are 70°, triangle BEF is isosceles. 55° 55° Alternate angles are equal Angles in a triangle add up 180° 180° - 70° - 55° = 55° As both base angles are 55°, triangle ABC is isosceles. 41° 113° 41 Interior angles add up to 180° 180° - 67° = 113° 113 2 4cm 3cm 3cm 4cm 6cm 2cm 2cm OTHER ANSWERS ALSO ALLOWED 6cm Perimeter of rectangle A 14cm =Perimeter of rectangle B = 16cm Difference = 16cm - 14cm 2 Perimeter is the length around a shape 6cm 4cm Perimeter is the length around a shape 4cm 6cm Perimeter of rectangle = 6cm + 4cm + 6cm + 4cm 20 Square has 4 equal sides 3cm for each side x x x x OTHER ANSWERS ALSO ALLOWED Because two lengths of 12cm makes 24cm which is more than the perimeter As evident from the rectangle drawn for part (a) 20cm 40cm 1cm 1cm 2cm 20cm 40cm 10cm 4cm 8cm 4cm 10cm 2cm 5cm 5cm 8cm Find the only rectangle which has a perimeter of 26cm 8 5 Kilo means a thousand 1km = 1000m 1000m Area = Length x Width = 1000m x 10m Rectangle A Split compound shape into two rectangles 200m Rectangle B Area of rectangle A = 100 x 30 Area of rectangle B = 200 x a 35 200a + 3000 = 10000 200a = 7000 Count the number of squares to find the area C B E Shaded Area = Area of square – Area of circle LEARN THE FORMULAE OFF BY HEART Area of square = length x width = 80cm x 80cm Area of circle = = 3.14 x 30cm x 30cm Area shaded = 6400 - 2826 3574 equal to less than (because the length around the shape is the same) (because more than half the rectangle is unshaded) 10cm 10cm Shaded Area = Area of big square ABCD – Area of the 4 congruent (identical) triangles Area of big square = length x width = 10cm x 10cm Area of one triangle = Area of one triangle = Area of one triangle = Area of four triangles = 82 Shaded area = Area of small square = 30cm x 30cm 900 Length of large square = 50 Area of floor = 300cm x 180cm Number of small tiles needed = Number of small tiles needed = 60 Perimeter B = 9cm Perimeter A = 10cm Perimeter is the length around a shape Perimeter C = 10cm Perimeter of D = 2cm + 2cm + 2cm + 2cm 8 A and C C and D Area is the space inside a shape Shaded Area = Area of big square – Area of two smaller squares Area of big square = length x width = 12cm x 12cm Area of one small square = length x width = 4cm x 4cm Area of both squares = Shaded Area = Area of big square = length x width Area of one small square = length x width Area of both squares = Unshaded Shaded Area = Fraction shaded = 6 A, B and E