1. Shape and Space 11. Similarity and Congruency
Mr F’s Maths Notes
Shape and Space
11. Similarity and Congruency

2. 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 IDENTICAL
They may have been flipped upside down and rotated around, but they are still exactly the same shape and the same size

3. 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

4. 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!

5. 4 cm 120o 35o 10 cm 8 cm 4 cm 120o 8 cm 13 cm 13 cm 5 cm 5 cm 20o
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
120o
35o
10 cm
8 cm
4 cm
120o
These two triangles are congruent because of AAS
8 cm
13 cm
13 cm
5 cm
5 cm
20o
12 cm
These two triangles are congruent because of RHS

6. 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!

7. C B A 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
To Find p:
18 cm
Okay, so we know the shapes are similar, so let’s work out the scale factor between rectangles A and B:
p cm
4 cm
q 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: B
48 cm A
16 cm
To Find q:
So now we have our scale factor, it’s dead easy to work out our missing length:
Okay, so now let’s work out how to get from Rectangle A to Rectangle C

8. 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
(a) How do you know these two triangles are similar?
(b) Find the unknown lengths
Part (a)
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…
3.4cm
350 X
2.5cm
250
1200
7.5cm Y
1200
6.3cm
To Find X
To Find Y
So, to get from one triangle to the other, we either multiply or divide by 3!

9. 60 cm ? 40 cm 20.25 litres 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
So, if our length scale factor = 1.5
Volume Scale Factor = = ?
40 cm
litres
60 cm
So now we know how to get from the big container to the small container, so we can work out its volume:

10. Good luck with your revision!