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CONGRUENT AND 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
In similar shapes:
• corresponding angles are equal
• corresponding sides are in the same ratio
C
D
A
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
Finding Missing Lengths in Similar Shapes
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
Starter
Page 127 Q5b
Find x and y
11
10
x
5
y
18
6
Example
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.
y
x
6
9
Using ratio of corresponding sides:
x 9

4 6
y 6

6 9
6x  36
9y  36
x6
y4
y
4
6
x
6
9
x
Question
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.
12
y
16
20
Using ratio of corresponding sides:
x 12

20 16
y 16

6 12
16x  240
12y  96
x  15
y 8
6
12
x
y
16
20
Page 126 Ex 3.11
Q1,2 and 9