Download Congruent Triangles

Congruent Figures
Figures are congruent if
they are exactly the same
size and shape.
These figures are
congruent because one
figure can be translated
onto the other figure.
Congruent Figures
Now look at these two figures:
They are also congruent
because one figure can be
rotated onto the other.
Congruent Figures
Finally look at these two figures:
They are also congruent because one figure
can be reflected onto the other figure.
Congruent Polygons
In general we can say that polygons are congruent if
the following two conditions are true:
•
The angles in one figure are the same as
the angles in the other figure and are in
the same order either clockwise or
anticlockwise.
•
The lengths of the sides between
corresponding pairs of adjacent angles
are the same for both figures.
Therefore to determine congruence we need to know
all of the angles and the lengths of all but one of
the sides. Not very practical!!
Congruent Triangles
As triangles have only three sides there are four
rules that allow us to determine whether two
triangles are congruent with only three pieces of
information:
•
(SSS) – If all of the sides on one
triangle are the same as the sides on
another then the triangles are
congruent.
3
5
2
2
5
3
Congruent Triangles
As triangles have only three sides there are four
rules that allow us to determine whether two
triangles are congruent with only three pieces of
information:
•
(SAS) – If two sides and the angle
between them are equal then the two
triangles are congruent.
4
20º
5
5
20º
4
Congruent Triangles
As triangles have only three sides there are four
rules that allow us to determine whether two
triangles are congruent with only three pieces of
information:
•
(AAS) – If two angles are equal and a side
on one triangle is the same length as a
corresponding side on the other triangle
then the triangles are congruent.
7
100º
20º
7
20º
100º
Congruent Triangles
As triangles have only three sides there are four
rules that allow us to determine whether two
triangles are congruent with only three pieces of
information:
•
(RHS) – If both triangles are right angled,
have equal hypotenuses and another equal
side then they are congruent.
6
2
6
2
Congruent Triangles
Before we prove congruence between triangles we
must practice our four congruent rules:
Eg.1
Are these two triangles congruent? If so,
which test confirms it?
B
D
F
80°
7
5
80°
70°
A
4
4
C
70°
E
ABC  EFD ( AAS)
Congruent Triangles
Before we prove congruence between triangles we
must practice our four congruent rules:
Eg.2
Are these two triangles congruent? If so,
which test confirms it?
B
7
5
80°
70°
A
D
4
F
4
C
70°
E
7
 ABC  EFD (SAS)
Congruent Triangles
To use the properties of congruence to help us solve
problems we first have to prove that one triangle is
congruent to another:
e.g.
Find the value of x in the diagram below giving
reasons for your answer:
B
D
F
80°
7
5
80°
70°
A
4
4
C
70°
E
x
ACB  FDE ( given)
BAC  DEF ( given)
AC  DE ( given)
ABC  EFD ( AAS)
 EF  AB
x  7
Congruent Triangles
e.g.
Find the value of x in the diagram below giving
reasons for your answer:
B
6
100°
AB  AD ( given)
4
BC  DC ( given)
C
A
AC (common side)
 ABC  ADC (SSS)
 ABC  ADC
6
xº
D
4
x  100