Similarity and Congruence
Introduction

Recognizing and using congruent and similar shapes can make calculations and design work easier. For instance, in the design below, only two different shapes were actually drawn. The design was put together by copying and manipulating these shapes to produce versions of them of different sizes and in different positions.



In this unit we will look in a little more depth at the mathematical meaning of the terms similar and congruent, which describe the relation between shapes like those in this design.
Congruence and similarity
The two shapes below are said to be congruent. This means that they are the same shape and size. If you move or rotate the shape on the right below, it will still be congruent to the shape on the left. The shapes would also remain congruent if you reflected the shape on the right, producing its mirror image, because all it sides and angles retain their size. Try moving and rotating the right-hand shape in Fig.1 below.


Click on the figure below to interact with the model.

 Figure 1.  Two congruent shapes.



Each box in Fig.2 below contains congruent shapes.

Figure 2.   Congruent shapes.
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The two shapes below are said to be similar. This means that they are the same shape but can be different sizes, i.e. they have the same proportions. If you move, rotate, reflect, or change the scale of the shape on the right in Fig.3 below, the two shapes will still be similar.


Click on the figure below to interact with the model.

 Figure 3.  Two similar shapes.



Each box in Fig.4 below contains similar shapes.

Figure 4.   Similar shapes.
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Which of the following statements is false?
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Complete this sentence using each word once.

  • All pairs of shapes are .
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The difference between congruence and similarity is that
similar
Two shapes are similar if one is congruent to an enlargement of the other. All squares are similar, as are all circles.
similar
shapes can be resized versions of the same shape, whereas congruent shapes must have identical lengths.

Figure 5.   Different squares.
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Think about all the ways that you might draw a square. A few examples are shown in Fig.5 above.

Which of the following statements is true?
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Figure 6.   Different triangles.
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Fig.6 shows some different triangles.

Which of the following statements is true?
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Figure 7.   Different circles.
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Finally, imagine all the circles that you could draw. Again, some different circles are shown in Fig.7 above.

Which of the following statements is true?
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Congruence
If two shapes are congruent, then we know that their measurements (including lengths and angles) are the same. The two triangles in Fig.8 below are congruent.

Figure 8.   Two congruent triangles.
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Look at Fig.8 and fill in the missing values in the spaces provided.

  • a =
    b = cm
    c =
    d =
    e =

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In congruent shapes, we say that equivalent side or
angle
An angle is a measure of turning. Angles are measured in degrees. The symbol for an angle is .
angle
pairs, such as angles e and d, are corresponding.

Figure 9.   Two congruent shapes.
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The two shapes in Fig.9 above are congruent. By sight, match up the corresponding pairs.
  • a
    b
    c
    d
    e
    f
    g
    h
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Similarity and proportion
As you can see in the unit Enlargement, Angle and Length,
enlargement
An enlargement is a type of transformation in which lengths are multiplied whilst directions and angles are preserved. The transformation is specified by a scale factor of enlargement and a centre of enlargement. For every point in the original shape, the transformation multiplies the distance between the point and the centre of enlargement by the scale factor.
enlargement
preserves the angles of a figure but multiplies each of its lengths by the same
constant
A constant is a quantity (such as a number or symbol) that has a fixed value, in contrast to a variable.
constant
(the
scale factor
The scale factor is the ratio of distances between equivalent points on two geometrically similar shapes.
scale factor
). So mathematically, any original shape and its enlarged image are similar.

The model in Fig.10 below shows a
triangle
A triangle is a three-sided polygon.
triangle
and its enlargement (on the right). The scale factor of enlargement is shown between them. You can type in a new scale factor of enlargement to see the second shape change size, or use the handle on the
image
A shape that is the result of a transformation on the coordinate plane.
image
.


Click on the figure below to interact with the model.

 Figure 10.  Two shapes related by enlargement.



Although the size of a shape changes during enlargement, it remains in proportion to the original shape. That is, although the new shape is a different size from the original, it retains the same proportions.

This is a useful aspect of similarity. In fact, it means that we can use whether shapes are in proportion to one another as a test for similarity. For instance, look at the two triangles in Fig.11 below (they are not necessarily drawn to scale). Marked on them are all the measurements that we know of them. Are they similar?

Figure 11.   Two triangles – are they similar?
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To find out, we first identify what would would be the corresponding sides if the triangles were similar.

In Fig.11 match up the sides, shortest to shortest and so on.
  • AB
    BC
    CA
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If these are indeed similar triangles, then the
ratio
A ratio compares two quantities. The ratio of a to b is often written a:b. For example, if the ratio of the width to the length of a swimming pool is 1:3, the length is three times the width.
ratio
of the lengths in each pair will be a constant (the scale factor of enlargement). We therefore need to calculate the ratio of lengths for each of the paired sides.

     So the triangles are related by an enlargement. They are therefore in proportion to one another and so they are similar triangles.



Are the two triangles in Fig.12 below similar?
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Figure 12.   Two triangles - are they similar?
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Summary

Congruent shapes are the same shape and size as one another, but can be in different positions, orientations, or form mirror images.

Shapes are similar if they have the same proportions. They may have different positions and orientations.

Whether shapes are congruent or similar can be found by comparing their internal angles and the length of their sides.
Exercises
Figure 13.   An assortment of shapes.
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1. Look at the shapes in the Fig.13 above and complete the following statements:

  • a) Shape 1 is congruent or similar to shapes ,, and .



    b) Shape 6 is congruent to shape .



    c) Shape 11 is to shape 12.

    d) Shapes 8 and 9 are to one another.

    e) Shape 5 is similar to shapes ,, and .
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Click on the figure below to interact with the model.

 Figure 14.  Two congruent trapezia

2. Look at the model in Fig.14 above. The two tapezia are congruent. What
transformation
A transformation on a shape is any operation which alters the appearance of the shape in a well defined manner.
transformation
applied to one of the shapes makes them similar, but not congruent?
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3. Choose the right word to complete the following statement. 'All equilateral triangles are ........ to each other.'
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Figure 15.   A cylinder and a cone.
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4. Look at the diagrams in Fig.15 above and complete these sentences.

  • Solid A can be thought of as a pile of thin, circles.

    Horizontal cross-sections of solid B would form a stack of circles.
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Figure 16.   Two triangles.
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5. Look at Fig.16 above and fill in the blanks in the statements below.

  • a) (corresponding angles on parallel lines)

    b) (corresponding angles on parallel lines)

    c) and therefore have the same set of angles. This means that they are triangles.
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