Mr Barton’s Maths Notes
Shape and Space
11. Similarity and Congruency
www.mrbartonmaths.com
With thanks to www.whiteboardmaths.com for the images!
11. Similarity and Congruency
1. If two shapes are Congruent, what does that mean?
When mathematicians say that two shapes are congruent, it is just a posh, complicated way of
saying that those shapes are IDENTTICAL
They may have been flipped upside down and rotated around, but they are still exactly the
same shape and the same size
2. Congruent Triangles
Because triangles only have three sides, and we know that all their interior angles must add up
to 1800, we don't actually need to know every single piece of information about two triangles
to be able to say that they are congruent (identical).
There are 4 sets of criteria, and if a pair of triangles match any of these, then we can say for
definite that they are the exact same triangle, and so they are congruent!
1. Three Sides equal (SSS)
The lengths of all three sides are given in the question, and they are the same for both triangles
2. Two Sides and the included Angle equal (SAS)
Two sides are the same length, and the angle in between those two sides is the same size!
3. Two Angles and a corresponding Side equal (AAS)
Two angles are equal, and so too is a side in the same position relative to those two angles!
4. Right angle, Hypotenuse and Side (RHS)
The triangle has a right angle, and you know the length of the hypotenuse and another side!
3. Examples
When answering questions on congruent triangles, you must quote one of the above four
conditions if you believe a pair of triangles to be congruent:
4 cm
35o
120o
10 cm
8 cm
4 cm
These two
triangles are
congruent because
of AAS
120o
8 cm
13 cm
13 cm
5 cm
20o
12 cm
5 cm
These two
triangles are
congruent because
of RHS
4. If two shapes are Similar, what does that mean?
• Unfortunately, when mathematicians says that two objects are similar, they do not mean
that they look a bit a like
• They mean that one object is an enlargement of the other
• Technically, to get from one object to the other you must multiply (or divide) every single
length by the same number
• Just like when we dealt with Enlargement, this number is called the Scale Factor!
5. Using Length Scale Factors
If we are told that two object are similar, and we can work out the scale factor, then it is
possible to work out a lot of unknown information about both objects
Example - These three shapes are similar. Find the missing values
18 cm
p cm
4 cm
A
To Find p:
Okay, so we know the
shapes are similar, so let’s
work out the scale factor
between rectangles A and B:
48  16  3
q cm
B
48 cm
16 cm
C
So, we must enlarge every
length on Rectangle A by a
scale factor of 3 to get the
lengths of Rectangle B.
So, our missing length must
be:
4  3  12cm
To Find q:
Okay, so now let’s
work out how to get
from Rectangle A to
Rectangle C
18  4  4.5
So now we have our
scale factor, it’s dead
easy to work out our
missing length:
16  4.5  72cm
6. Similar Triangles
For any other shape to be similar, all angles must be the same and all matching sides must be in
proportion
But… because triangles are funny, all you need for similarity between two triangles is for all
three angles to be the same. Then you can be sure one triangle is an enlargement of the other
Example
Part (a)
(a) How do you know these two triangles are similar?
(b) Find the unknown lengths
3.4cm
350
X
2.5cm
250
7.5cm
Y
1200
1200
6.3cm
To Find X
3.4  3  10.2cm
To Find Y
6.3  3  2.1cm
Two triangles are similar if all
their angles are the same…
Well… if you work out the
missing angle in the yellow
triangle it is 250, and the missing
angle in the green triangle is…
350
So… all the angles are the same,
so the triangles are similar!
And because they are similar, we
can work out the scale factor,
using our matching sides between
the 1200 and the 350…
7.5  2.5  3
So, to get from one triangle to
the other, we either multiply or
divide by 3!
7. Area and Volume Factors
It is also possible for 3D shapes to be similar.
If we can work out the scale factor between their lengths of sides, we can also say that:
Area Factor
= Scale Factor2
Volume Factor = Scale Factor3
Okay, before we can do anything
we need to work out the length
scale factor in exactly the same
way as we always do:
Example - These two containers are similar. Work out
the volume of water the smaller one can hold
60  40  1.5
?
40 cm
20.25
litres
60 cm
So, if our length scale factor = 1.5
Volume Scale Factor = 1.53 = 3.375
So now we know how to get from
the big container to the small
container, so we can work out its
volume:
20.25  3.375  6 litres
Good luck with
your revision!