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KS3 Mathematics

KS3 Mathematics. S2 2-D shapes. S2 2-D shapes. Contents. S2.2 Polygons. S2.1 Triangles and quadrilaterals. S2.3 Congruence. S2.4 Tessellations. S2.5 Circles. Labelling lines and angles in a triangle. The side opposite A is called side a. The side opposite B is called side b.

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KS3 Mathematics

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  1. KS3 Mathematics S2 2-D shapes

  2. S2 2-D shapes Contents S2.2 Polygons S2.1 Triangles and quadrilaterals S2.3 Congruence S2.4 Tessellations S2.5 Circles

  3. Labelling lines and angles in a triangle The side opposite A is called side a. The side opposite B is called side b. The side opposite C is called side c. This triangle can be described as ABC. When we label angles in a triangle we use capital letters to label the vertices, going round in order, clockwise or anticlockwise. A b c C B a

  4. Labelling lines and angles in a triangle PQR How can we complete the labeling of this triangle? P r q ? Q ? R ? p This triangle is called

  5. Right-angled triangle A right-angled triangle contains a right angle. Triangles are named according to their properties. The longest side opposite the right angle is called the hypotenuse.

  6. Isosceles triangle Triangles are named according to their properties. An isosceles triangle has two equal sides. The equal sides are indicated by these lines. Lines of symmetry are indicated with dotted lines. The two base angles are also equal. An isosceles triangle has one line of symmetry.

  7. Equilateral triangle Triangles are named according to their properties. An equilateral triangle has three equal sides and three equal angles. An equilateral triangle has three lines of symmetry and has rotational symmetry of order 3.

  8. Scalene triangle Triangles are named according to their properties. A scalene triangle has no equal sides and no equal angles. A scalene triangle does not have any lines of symmetry.

  9. Naming triangles What type of triangle is this? This is a right-angled isosceles triangle. What symmetry properties does it have? It has one line of symmetry and no rotational symmetry.

  10. Triangles on a 3 by 3 pegboard

  11. Quadrilaterals Equal and/or parallel sides Equal angles Right angles Diagonals that bisect each other Diagonals that are at right angles Line symmetry Rotational symmetry Quadrilaterals are also named according to their properties. A shape can be classified according to whether it has:

  12. Parallelogram In a parallelogram opposite sides are equal and parallel. The diagonals of a parallelogram bisect each other. A parallelogram has rotational symmetry of order 2.

  13. Rhombus A rhombus is a parallelogram with four equal sides. The diagonals of a rhombus bisect each other at right angles. A rhombus has two lines of symmetry and it has rotational symmetry of order 2.

  14. Rectangle A rectangle has opposite sides of equal length and four right angles. A rectangle has two lines of symmetry.

  15. Square A square has four equal sides and four right angles. It has four lines of symmetry and rotational symmetry of order 4.

  16. Trapezium A trapezium has one pair of opposite sides that are parallel. Can a trapezium have any lines of symmetry? Can a trapezium have rotational symmetry?

  17. Isosceles trapezium In an isosceles trapezium the two opposite non-parallel sides are the same length. The diagonals of an isosceles trapezium are the same length. It has one line of symmetry.

  18. Kite A kite has two pairs of adjacent sides of equal length. The diagonals of a kite cross at right angles. A kite has one line of symmetry.

  19. Arrowhead An arrowhead or delta has two pairs of adjacent sides of equal length and one interior angle that is more than 180°. Its diagonals cross at right angles outside the shape. An arrowhead has one line of symmetry.

  20. Quadrilateral family tree QADRILATERAL KITE TRAPEZIUM PARALLELOGRAM RHOMBUS RECTANGLE ISOSCELES TRAPEZIUM DELTA SQUARE This family tree shows how the quadrilaterals are related.

  21. True or false

  22. Guess the shape It is a quadrilateral. Its diagonals cross at right angles. It has one vertical line of symmetry. It has two pairs of adjacent sides of equal length. None of its interior angles are more then 180°. Can you identify this shape from its properties? The shape is a kite.

  23. Guess the shape It is a quadrilateral. Its diagonals are of equal length. It has one vertical line of symmetry It has one pair of parallel sides. One pair of opposite, non-parallel sides are the same length. Can you identify this shape from its properties? The shape is an isosceles trapezium.

  24. Quadrilaterals on a 3 by 3 pegboard

  25. Shapes on a circle pattern

  26. S2 2-D shapes Contents S2.1 Triangles and quadrilaterals S2.2 Polygons S2.3 Congruence S2.4 Tessellations S2.5 Circles

  27. Tangram puzzle

  28. Polygons A polygon is a 2-D shape made when line segments enclose a region. A The end points are called vertices. One of these is called a vertex. B The line segments are called sides. E C D 2-D stands for two-dimensional. These two dimensions are length and width. A polygon has no height.

  29. Naming polygons Polygons are named according to the number of sides they have. Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon

  30. Polygons A regular polygon has equal sides and equal angles. In a convex polygon all of the interior angles are less than 180°. In a concave polygon some of the interior angles are more than 180°. All regular polygons are convex.

  31. Identify the polygon Which of the following are polygons?

  32. Overlapping squares

  33. Pentagon problem The interior angle of a regular pentagon is 108°. A Two diagonals are drawn from vertex A to make 3 triangles. E B 108° What triangles do they make? What properties do they have? D C Calculate the size of the angles in each triangle.

  34. Pentagon problem DAE and ADE are equal because they are the base angles in an isosceles triangle. DAE = ADE = Using the symmetry of a regular pentagon we can fill in the angles in ABC. A Using the fact that the angles in a triangle add up to 180° we can deduce that 36° 36° 36° E B 108° 108° 36° 36° 36° 72° 72° D C We can deduce the remaining angles from the fact that each interior angle in a regular pentagon equals 108°.

  35. S2 2-D shapes Contents S2.1 Triangles and quadrilaterals S2.2 Polygons S2.3 Congruence S2.4 Tessellations S2.5 Circles

  36. Congruent shapes B P C A R Q If shapes are identical in shape and size then we say they arecongruent. Congruent shapes can be mapped onto each other using translations, rotations and reflections. These triangles are congruent because AB = PQ, BC = QR, and AC = PR. A =  P, B = Q, and C = R.

  37. Similar shapes Q B C R A P If one shape is an enlargement of the other then we say the shapes are similar. Two similar shapes have the same angle sizes but different side lengths. The corresponding side lengths of two similar shapes are always in the same ratio. These triangles are similar because A =  P, B = Q, and C = R. AB:PQ = BC:QR = AC:PR

  38. S2 2-D shapes Contents S2.1 Triangles and quadrilaterals S2.2 Polygons S2.4 Tessellations S2.3 Congruence S2.5 Circles

  39. Tessellations If a shape can completely fill a flat space without any gaps or overlaps then it is said to tessellate. For example, this T-shape can be used to produce the following tessellation:

  40. Tessellating regular polygons Which regular polygons tessellate? Equilateral triangles? Squares?

  41. Tessellating regular polygons Which regular polygons tessellate? Regular pentagons? ? Regular hexagons?

  42. Tessellating regular polygons Explain why no other regular polygons tessellate. Semi-regular tessellations can be produced by combining two or more regular polygons. For example, combining squares with regular octagons produces the following semi-regular tessellation:

  43. Tessellating shapes Decide which of the following shapes will tessellate. If shapes won’t tessellate can you give a reason why?

  44. Tiling patterns Look at this tiling pattern. How would you describe the pattern to someone over the phone?

  45. Tiling patterns Look at this tiling pattern. Suppose we wanted to make this pattern to tile a bathroom floor. How could the pattern be divided into square tiles?

  46. Tiling patterns Here is one of the square tiles used to make the pattern. Can you see how this tile could be constructed? We can start with an 8 by 8 square grid. Construct squares inside the grid as follows … … and then colour.

  47. Tiling patterns Rotations and reflections of this tile can be used to make many different tiling patterns. For example, or

  48. Tiling patterns This tiling pattern can be found in the Alhambra palace in Granada, Spain. Can you see a way to divide this pattern into square tiles?

  49. Investigating tiling patterns

  50. S2 2-D shapes Contents S2.1 Triangles and quadrilaterals S2.2 Polygons S2.5 Circles S2.3 Congruence S2.4 Tessellations

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