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SIMILAR TRIANGLES
Congruency
Two shapes are congruent if one of the shapes fits exactly on top of the other shape.
These three triangles
are all congruent
In congruent shapes:
• corresponding angles are equal
• corresponding lengths are equal
To prove that two triangles are congruent you must show that they satisfy one
of the following four sets of conditions:
SSS: three sides are equal
SAS: two sides and the included
angle are the same
ASA: two angles and the included
side are the same
RHS: right-angled triangle with
hypotenuse and one other side the
same
Which triangles are congruent to triangle A?
A
Similar shapes
R
These two quadrilaterals are similar.
S
PQ QR RS SP



AB BC CD DA
C
D
In similar shapes:
• corresponding angles are equal
A
• corresponding sides are in the same ratio
To show that two triangles are similar it is sufficient
to show that just one of the above conditions is satisfied.
B
P
Q
Which triangles are similar to triangle A?
A
Examples
1 The triangles are similar. Find the values of x and y.
(All lengths are in cm.)
x
5.5
y
9
8
12
Using ratio of corresponding sides:
x
12

5.5 8
y
8

9 12
8x  66
12y  72
x  8.25
y6
Examples
2 The triangles are similar. Find the values of x and y.
(All lengths are in cm.)
7
Turn one of the triangles so
that you can see which are
the corresponding sides.
x
8
7
Using ratio of corresponding sides:
y
15
12
x
8

15 12
12x  120
x  10
y 12

7 8
8y  84
y  10.5
Examples
3 Find the values of x and y.
(All lengths are in cm.)
x
6
Separate the two triangles.
9
2.5
2
Using ratio of corresponding sides:
y
y 8

9 6
x  2.5 8

x
6
6y  72
6(x  2.5)  8x
y  12
6x  15  8x
x +2.5
8
y
2x  15
x  7.5
x
6
9
6
Examples
4 Find the values of x and y.
(All lengths are in cm.)
4
Turn the top triangle so that you can
see which are the corresponding
sides.
Using ratio of corresponding sides:
x 9

4 6
y 6

6 9
6x  36
9y  36
x6
y4
y
x
6
9
y
4
6
x
6
9
x
Examples
5 Find the values of x and y.
(All lengths are in cm.)
6
Turn the top triangle so that you can
see which are the corresponding
sides.
Using ratio of corresponding sides:
x 12

20 16
y 16

6 12
16x  240
12y  96
x  15
y 8
12
y
16
20
6
12
x
y
16
20