Congruence
Wackford Squeers
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Congruence Describe what is meant by Congruence. Solve problems involving congruence
Definition Congruent means exactly the same as. So two triangles are congruent if one can be placed exactly on top of the other – you are allowed to rotate it and even flip it over to check that it fits
All the above triangles are congruent
Proving Two Triangles are Congruent We need to prove that certain parts of two triangles are the same – and this will prove that the whole triangles are congruent The rules for congruent triangles can be written quite simply:
SAS – this means you prove that Side, Angle, Side are the same SSS – this means proving all three sides are the same ASA – this means proving two angles and the included side are the same We can see these rules illustrated as follows:
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Notice: that we have two lines – and the
Notice: in this case we have all 3 sides – we don’t need the angles (SSS)
Notice: in this case we have one side – and INCLUDED two angles angle (SAS) coming off it (ASA) Now that we have the ‘rules’ we can see about how to use them:
Worked Example 1 Let’s have a look at an Isosceles triangle with a perpendicular drawn from the top to the base. It looks like this
A
B
θ
θ D
C
We are going to look at the two triangles ABD and ADC to see if we can prove that they are congruent Line AB = AC (sides of an Isosceles triangle) AD is a line common to both triangles CAD = BAD (each triangle has a right-angle and the base angles are equal – therefore these two angles are equal) So, we have SAS (AD DAC and AC are the same as BA BAD and AD) So, the triangles are congruent
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Worked Example 2 A parallelogram with a diagonal drawn looks like this:
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B
D
C
We’re going to prove that triangle ADB is congruent to triangle DBC
A
B
D
B
C
D
DB is common to both triangles BC = AD (opposite sides of a parallelogram) AB = DC (opposite sides of a parallelogram) So, we have three sides the same – SSS triangle ADB triangle DBC
Exercise 1
B
C
D
A
F
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ABCDEF is a regular hexagon Prove that triangle ABC triangle BCD
Exercise 2 B
C
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A
F
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Prove that triangle BFE triangle BEC
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