Congruent and similar shapes - KS3 Maths

Key points

A diagram of a rectangle reflected in a mirror line. The dimensions of the original rectangle and the reflected image are identical. — Image caption,
These reflected rectangles are example of congruent shapes.

When transforming a shape, either through translation, reflection or rotation, a congruent shape is produced. Enlargements create shapes that are similar.
When two shapes are the same in shape and size, they are shapes. When two shapes are different in size but proportionally the same shape, they are shapes.
To understand congruence and similarity, a good understanding of the properties of polygons, including triangles and quadrilaterals, can be helpful.

Congruent shapes

Two shapes are described as if they are identical.

The lengths of sides (edges) and sizes of angles must be equal between the two shapes for them to be congruent.
or change the orientation of a shape but they are still congruent to the original shape.
A great way to check if two shapes are congruent is to place one on top of the other. If they match, with no overlaps, they are congruent.

Examples

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Two of these triangles are congruent, which means they are identical in shape and size.
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Triangles ABC and GHI are congruent. Triangle GHI is the same as ABC but has been rotated 90° anticlockwise. Triangle DEF is a different size.
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Line segments AB and GH are the same length. BC and HI are the same length. AC and GI are the same length.
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Angle A and angle G are equal. Angle B and angle H are equal. Angle C and I are equal and are both right angles.
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These two rectangles are not congruent.
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Rotating rectangle EFGH and placing it on top of rectangle ABCD shows they are not the same size.
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These two quadrilaterals are congruent.
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EFGH is a reflection of ABCD on a vertical mirror line. Line segments AB and EH are the same length. They correspond to each other, which means they are in the same place in two different shapes. BC and GH are the same length. CD and FG are the same length. AD and EF are the same length.
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Angles in congruent shapes are equal. Angle A and angle E are equal. They correspond to each other. Angle B and angle H are equal. Angle C and G are equal. Angle D and F are equal.

Question

Look at the grid of shapes. How many congruent pairs can you find?

An image of a rectangle which has been split into thirteen shapes of various types and sizes. Shapes A, E, J and K are rectangles. Shapes B, D and M are squares. Shape C is an L shape. Shape F is a trapezium. Shapes G, H, I and L are triangles.

Similar shapes

Two shapes are described as if one is an of the other. The shapes do not need to be the same way.

The sizes of angles must be equal between the two shapes.
The shapes must also be the same. If one side on the enlarged shape doubles in length, all sides must be double the original size shape. The increase in size from one shape to another is called a .
The shapes may need to be to find out if they are similar.

It is possible to calculate missing lengths on similar shapes when given either the scale factor or enough information to calculate it.

Examples

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All of these rectangles are different sizes, but two are proportionally the same and similar.
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Rotating rectangle EFGH makes it easier to visualise which two shapes are similar. Rectangles EFGH and IJKL are similar. Their lengths are three times their widths. ABCD is too short in length to be similar.
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Rectangle IJKL is an enlargement by scale factor two. All dimensions have doubled in length. Line EF is 1 cm and line IL is 2 cm. Line EH is 3 cm and line KL is 6 cm. This means EFGH is half the scale of IJKL. All the angles are the same (90°).
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Isosceles triangles ABC and DEF are not proportionality the same. The base in EF is twice the size as BC. EF ÷ BC = 6 ÷ 3 = 2. The sides DE and DF are three times the size AB and AC. DF ÷ AC = 12 ÷ 4 = 3. This means triangles ABC and DEF are not similar.
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Triangles ABC and DEF are similar. There is enough information to work out the scale factor.
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By comparing any pair of corresponding sides, eg DE and AB, the scale factor can be calculated. Scale factor = DE/AB. Scale factor = 15/10. Scale factor = 1·5. Triangle DEF is an enlargement by scale factor 1·5
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Angle A and angle D are equal. Angle B and angle E are equal. Angle C and F are equal and are both right angles.
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Rectangles ABCD and EFGH are similar. Rectangle EFGH is an enlargement scale factor 3. Calculate the length of GH.
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All dimensions have multiplied by 3. GH = CD × 3. GH= 1·8 × 3. GH = 5·4 m.

Question

Which of these two triangles are similar?

A series of three images. Each image shows a different size right angled triangle. The first triangle has vertices labelled A, B, and C. A B has length five centimetres. B C has length four centimetres and C A has length three centimetres. The right angle is at vertex C. The second triangle has vertices labelled D, E, and F. D E has length eight centimetres. E F has length six centimetres and F D has length ten centimetres. The right angle is at vertex E. The third triangle has vertices labelled G, H, and I. G H has length twelve centimetres. H I has length five centimetres and I G has length thirteen centimetres. The right angle is at vertex H.

Practise working with congruent and similar shapes

Quiz

Practise working with congruent and similar shapes with this quiz. You may need a pen and paper to help you with your answers.

Real-life maths

An image of bees on a beehive structure.

Some congruent shapes can fit together without gaps. If this is possible the shape is said to tessellate.

The construction of a beehive is a real-life example of a tessellation pattern.

The hexagon shapes fit together to form the structure. The bees use this pattern as it is an efficient use of the space which involves as little wax, the material which they use, as possible.

Similar tessellation patterns can be found in the way paths, driveways and brick walls are designed.