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**KS3 Mathematics S2 2-D shapes**

The aim of this unit is to teach pupils to: Identify and use the geometric properties of triangles, quadrilaterals and other polygons to solve problems; explain and justify inferences and deductions using mathematical reasoning Understand congruence and similarity Identify and use the properties of circles Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp S2 2-D shapes

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**S2.1 Triangles and quadrilaterals**

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

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**Labelling lines and angles in a triangle**

When we label angles in a triangle we use capital letters to label the vertices, going round in order, clockwise or anticlockwise. The side opposite A is called side a. A b c The side opposite B is called side b. C Introduce the labelling conventions for triangles. Tell pupils that in a triangle, the angle opposite the longest side is the largest angle. The angle opposite the shortest side is the smallest angle. The side opposite C is called side c. B a This triangle can be described as ABC.

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**Labelling lines and angles in a triangle**

How can we complete the labeling of this triangle? P r q ? Q p ? R ? This triangle is called PQR

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**Right-angled triangle**

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

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**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.

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Equilateral triangle Triangles are named according to their properties. An equilateral triangle has three equal sides and three equal angles. An equilateral triangle is defined as having three equal sides. The three angles are equal as a consequence of this definition. Ask pupils to tell you the size of the three equal angles (180º ÷ 3). An equilateral triangle has three lines of symmetry and has rotational symmetry of order 3.

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**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.

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**Naming triangles What type of triangle is this?**

This is a right-angled isosceles triangle. This slide shows that a triangle can be both right-angled and isosceles. Ask pupils to tell you the size of all the angles in a right-angled isosceles triangle. Ask pupils: is it possible to have a right-angled equilateral triangle? (no); A right-angled scalene triangle? (yes) What symmetry properties does it have? It has one line of symmetry and no rotational symmetry.

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**Triangles on a 3 by 3 pegboard**

Challenge pupils to find the eight distinct triangles (not including reflections, rotations and translations) that can be made on a 3-by-3 pegboard. Classify them according to their side, angle and symmetry properties. There should be 2 scalene, 3 right-angled isosceles, 1 right-angled and 2 isosceles triangles.

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Quadrilaterals Quadrilaterals are also named according to their properties. A shape can be classified according to whether it has: equal and/or parallel sides equal angles right angles diagonals that bisect each other List the properties that we use to classify shapes. Some of these properties define the shape. These are the minimum requirements needed to define the shape. Other properties, such as symmetry properties, are derived properties and arise as a result of the definition. diagonals that are at right angles line symmetry rotational symmetry.

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**Parallelogram In a parallelogram opposite sides are equal**

and parallel. Draw pupils’ attention to the convention of using double dashes to distinguish between the two pairs of equal sides and the use of double arrow heads to distinguish between two pairs of parallel sides. State that when two lines bisect each other, they cut each other into two equal parts. Ask pupils for other derived properties such as the fact that the opposite angles are equal and adjacent angles add up to 180º. Ask pupils if they know the name of a parallelogram that has four right angles (a rectangle), a parallelogram that has four right angles and four equal sides (a square) and a parallelogram with four equal sides (a rhombus). The diagonals of a parallelogram bisect each other. A parallelogram has rotational symmetry of order 2.

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**Rhombus A rhombus is a parallelogram with four equal sides.**

Ask pupils for other derived properties such as the fact that the opposite angles are equal. Ask pupils if they know the name of a rhombus that has four right angles (a square). 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.

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**Rectangle A rectangle has opposite sides of equal length**

and four right angles. Ask pupils for other derived properties. For example, the diagonals are of equal length and bisect each other. Ask pupils to explain why it is possible to describe a square as a special type of parallelogram. (Because a square has all the properties of a parallelogram.) A rectangle has two lines of symmetry.

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**Square A square has four equal sides and four right angles.**

Ask pupils for other derived properties such as the fact that the diagonals are of equal length and bisect each other at right angles. Ask pupils to explain why it is possible to describe a square as a special type of parallelogram, a special type of rhombus or a special type of rectangle. and rotational symmetry of order 4. It has four lines of symmetry

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Trapezium A trapezium has one pair of opposite sides that are parallel. A trapezium has one line of symmetry when the pair of non-parallel opposite sides are of equal length. No trapezium has rotational symmetry. Can a trapezium have any lines of symmetry? Can a trapezium have rotational symmetry?

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Isosceles trapezium In an isosceles trapezium the two opposite non-parallel sides are the same length. Ask pupils for other derived properties such as the fact that there are two pairs of equal adjacent angles. The diagonals of an isosceles trapezium are the same length. It has one line of symmetry.

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**Kite A kite has two pairs of adjacent sides of equal length.**

Ask pupils for other derived properties such as the fact that there is one pair of opposite angles that are equal. Ask pupils if a kite can ever have parallel sides. Conclude that this could only happen if the four sides were of equal length, in which case it would no longer be a kite, but a rhombus. The diagonals of a kite cross at right angles. A kite has one line of symmetry.

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Arrowhead An arrowhead or delta has two pairs of adjacent sides of equal length and one interior angle that is more than 180°. Ask pupils for other derived properties such as the fact that one pair of angles is equal. Ask pupils if a kite can ever have parallel sides. (No.) Its diagonals cross at right angles outside the shape. An arrowhead has one line of symmetry.

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**Quadrilateral family tree**

This family tree shows how the quadrilaterals are related. QUADRILATERAL KITE PARALLELOGRAM TRAPEZIUM Use this family tree to explain how the quadrilaterals are related. For example, a rectangle is a special type of parallelogram with four right angles. A square is a special type of rectangle, rhombus and parallelogram. All of the shapes are quadrilaterals (at the top of the tree). RHOMBUS RECTANGLE ISOSCELES TRAPEZIUM DELTA SQUARE

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True or false Discuss each statement with the class and ask the pupils to say whether they think each statement is true or false. If pupils cannot agree, ask them to give examples or counter-examples to support their argument.

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**Can you identify this shape from its properties?**

Guess the shape Can you identify this shape from its properties? 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 than 180°. As each property is revealed ask pupils to tell you what quadrilaterals could have this property. ? The shape is a kite.

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**Can you identify this shape from its properties?**

Guess the shape Can you identify this shape from its properties? 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. As each property is revealed ask pupils to tell you what quadrilaterals could have this property. ? The shape is an isosceles trapezium.

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**Quadrilaterals on a 3 by 3 pegboard**

Challenge pupils to find the 16 distinct quadrilaterals (not including reflections, rotations and translations) that can be made on a 3-by-3 pegboard. Classify them according to their side, angle and symmetry properties.

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