Congruent and Similar Figures Booklet
PART 1: CONGRUENCY
Congruent Figures: Congruent figures are
IDENTICAL. All angles are the same. All
sides are the same.
However, congruent triangles may look
different if they are ROTATED or
REFLECTED.
NOTATION:
The symbol for congruent is
with a squiggle on top).
 (an equal sign
When we say things are congruent, we have to
use this symbol and we have to keep the
letters in the order that they correspond
(match up).
We can say the following about the diagrams
above:
LMNO
ABC


Practice 1) Which of the shapes below are
congruent to A?
Write statements here using the symbol
A
A
A



:
PART 2: SIMILARITY
Enlargements, reductions, and scale drawings
are all examples of similar figures.
Two figures are similar if they have the
same shape but not necessarily the same
size.
Triangles ABC and XYZ below are similar. If
triangle ABC was reduced in size it would look
exactly like XYZ. But triangle DEF is not
similar to the others it has a different shape.
B
Y
E
A
C
D
F
X
NOTATION:
The symbol for similar is ~ (a squiggly line).
Z
Practice 2) Which of the shapes below are
similar to A?
Write statements here using the symbol ~ :
A~
- The similarity of the two figures depends
on the correspondence – the way the parts
are matched up.
Example 1: Triangle ABC is similar to triangle
XYZ. Give the corresponding parts.
Corresponding angles
 A~ X
 B ~ Y
C ~Z
Corresponding sides
AB ~ XY
Since A corresponds to X
and B corresponds to Y, side
AB corresponds to XY
AC ~ XZ
BC ~ YZ
The corresponding parts are given by the
order of the letters in the similarity
statement.
Triangle A B C is similar to triangle X Y Z.
Similarity Statement
 ABC ~  XYZ
Sometimes you might have to imagine “moving”
one of the two similar figures to see the
corresponding parts.
Example 2: Give the corresponding parts of
theses similar figures
R
G
M
L
W
T
Give the smaller triangle first a “turn” then a
“flip”
R
M
M
Turn
Flip
M
G
L
G
G
L
W
L
Now it is easy to write a similarity statement
and give the corresponding parts
T
Triangle GLM is similar to triangle WTR
The symbol  means
GL ~ WT
 G ~ W
“angle”
 L ~ T
M ~R
LM ~ TR
GM ~ WR
Exercises
Give the letter of the one figure that looks
similar to the first figure
1)
a.
b.
c.
a.
b.
c.
a.
b.
c.
a.
b.
c.
2)
3)
4)
Give the corresponding angle or side. Figure
ABCD is similar to figure RSWT
Corresponding angles Corresponding
sides
5)
 A~?
9)
6)
B~?
10)
7)
D~?
11)
8)
? ~ W
12)
W
C
AB ~ ?
T
D
AD ~ ?
A
B
? ~ WT
S
R
BC ~ ?
13) Triangle DEF is similar to triangle MNP.
Give the
3 pairs of corresponding angles and the 3
pairs of
F
corresponding sides
P
E
M
D
N
14) Triangle XYZ is similar to triangle QRS.
Give the 3 pairs of corresponding angles
And the 3 pairs of corresponding sides
Z
S
Y
X
R
Q
Write a similarity statement and give all pairs
of corresponding sides.
C
15)
16) J
F
I
P
A
B
U
17)
E
Z
W
M
G
T
S
R
D
18)
X
Y
O
N
H
T
C
G
A
D
O
K
Q
A
P
Z
19)
Y
20)
M
O
E
D
F
X
Z
C
21)
A
22)
D
E
B
X
P
Hint for Exercises 21 and 22
Q
Y
Imagine “sliding” the
smaller triangle off the
larger one. Notice that
 B is in both triangles. It
corresponds to itself
Give all pairs of corresponding angles and
sides.
23) Triangle PQR is similar to triangle MTF.
24) Triangle YWZ is similar to triangle PEG.
25) Triangle DEF is similar to triangle KLO.
26) Triangle TUV is similar to triangle ABC.
You know that similar figures are those that
have the same shape, but here is a more
precise description
Similar figures are those in which:
1) Corresponding angles have the
same measure (ie, corresponding
angles are equal)
2) Corresponding lengths are in
the same ratio
Example 1:
Z
C
B
A
X
Y
By measuring the angles with a protractor,
you can see that corresponding angles have
the same measure.
Angle Measure
Angle Measure
A
90
X
90
B
37
Y
37
C
53
Z
53
By counting units of the grid you can see that
corresponding lengths are in the same ratio.
AB 4 1
 
XY 8 2
each ratio 
AC 3 1
 
XZ 6 2
BC
5 1


YZ 10 2
length from  ABC
1

correspond ing length of XYZ 2
Example 2: Triangle MNP is similar to
triangle TUV. Find all the missing parts.
V
“a” stands for the
31 
length of side MP
P
24
x
a
“x” stands for the
length of side
UV
M
8
24
125 
10
N
T
15
U
There are 5 missing parts.
1)  P :  P corresponds to  V , so  P is 31
2)  T :  T corresponds to  M , so  T is 24
3)  U :  N corresponds to  U , so  U is 125
4) MP : The ratio of corresponding lengths are
the same so a proportion can be set up
Notice that corresponding letters
are above and below one
another
MN MP

TU
TV
Substitute the know lengths
10 a

15 24
Take the cross-products
10  24  15  a
240  15a
Divide both sides by 15
240 15a

15
15
16  a
So MP is
16.
5)
: Again use proportion.
UV
MN NP

TU UV
Substitute
10 8

15 x
Take Cross-products
10 x  120
Divide
is 12.
x  12
So UV
Exercises:
The figures
in Zeach pair are similar.
Use
C
L
12
6
proportion to
the missing sides
10 find
x
K
18
M
b
1)
X
A
2)
Y
6
15
D
B
9
c
E
10
C
d
V
F
G
Z
y
12
9
x
39
24
x
w
E
D
6
H
I
8
3)
X
Y
10
4)
T
15
U
Find all the missing parts of the similar
figures
C
5)
A
16
F
24
46 
10
D
29 
c
B
105 
20
w
E
5 missing parts
Y
30
V
b
83 
T
6)
57
X
16
20
y
20
40 
5 missing parts
Z
U
S
60 
N
56
b
M
a
R
42
120 
8
7)
K
90 
90 
28
14
P
L
Q
x
7 missing parts
D
C
24
90 
U
127 
40
32
T
15
w
z
8)
A
90 
n
B
R
53 
30
S
7 missing parts
Indirect Measurement
Similar figures and proportions can be used to
find measurements when it is difficult to
measure directly.
Example 1: A 3m pole has a 2m shadow. At
the same time a tree has a 10 m
h
shadow. How tall is the tree?
10m
3m
2m
This sketch shows the two
similar triangles. You can use
a proportion
2 3

10 h
2h  30
products
h  15
Take cross
Divide by 2.
The tree is 15m tall.
The pole and its shadow were a ready made
similar figure. If there is no such figure, you
can draw one from measurements. Straight
distances can be measured with a meter stick.
Angles can be measured with an instrument
such as a transit, a device used by surveyors.
Example 2: From a point on the shore of a
tree
river directly across from a tree you
walk 20 m along the shore. The tree
w
42
90 
is now at a 42° angle with the shore.
20m
Original
New
How wide is the river?
point
Place
Make a scale Tdrawing of any size
from the given measurements
O
90 
42
N
ON was drawn 4cm
long. Therefore the
scale of the drawing is:
4cm=20m
Now that the scale drawing is finished you
can write a proportion. The distance to be
found is the width w of the river. That
corresponds to OT on the scale drawing.
Carefully measure OT . It is 3.6cm.
4 3 .6

20
w
4w  72.0
The river is 18m wide.
x  18.0
EXERCISES
Triangle ABC is similar to triangle DEF. Copy
C
and complete each proportion
1)
AB BC

DE
?
4)
AC
?

DF EF
2)
BC AC

EF
?
3)
5)
DE EF

AB
?
6)
7) How tall is the tree?
h
2m
1.5m
4.5m
A
BC
?

EF DE
DE DF

AB
?
D
B
F
E
8) How high is the tip of the power pole?
h
1.7m
2m
40m
Express your answer to one decimal place in
each of the following problems.
D
1) To find the height of a tree, h in
metres
A
use the following diagrams.
1.9m
E
B
3.8m
h
18.0m
F
C
a) Which triangles are similar?
b) Write an equation to solve for h
c) Solve the equation in (b)
d) What is the height of the tree?
C
20.8m B
2) The width, w in metres, of a channel
is8.2m
19.6m
D
shown in the diagram.
E
a) Which triangles are similar?
b) Write the equation to solve for w
c) Solve the equation in (b)
d) What is the width of the channel?
w
A
3) To calculate the length of a lake in metres,
y
measurements are recorded onP the diagram.
18.6m
a) Which triangles are similar?
3.6m
b) Write the equation
to16.2m
solveU for y
V
c) Solve the equation in (b).
d) What is the length of the lake?
T
For each of the following problems, sketch a
copy of the diagram. Record the given
information. Solve the problem.
4) To calculate the height of a tree,
AB, the
B
E
following measurements were
made.
h
CD = 3.0m
AD = 12.0m
DE = 1.0 m
C
D
Calculate the height of the tree
A
S
5) In the diagram, PQ represents the width
P
of a river. Use the
measurements to find the
width of the river.
34.2m B
Q
79.6m
C
18.6m
A
x
B
D
6) To calculate the length of a trout pond, the
A
C
E
following measurements were made.
AB = 17.1m
AC = 15.2m
CE = 39.8m
Use the diagram. Calculate the length of the
pond.
7) A mirror is placed on the ground to
calculate the height of a building. Jennifer
places the mirror so that she sees the
reflection at the top of the building.
x
1.8m
1.2m
mirror
12.8m
Use the diagram. Calculate the height of the
building.
8) In a camera, similar triangles occur as
shown. Use the information in the diagram.
Calculate the height of the tree.
Image in
camera
Actual Tree
4.2 cm
3.8 cm
12.8 cm
x
9) On a sunny day, John’s shadow is 2.9m long,
while the shadow of a tower is 11.3m long. If
John is 1.8m tall, calculate the height of the
tower.
10) The shadow of a metre stick is 2.7m long,
while the shadow of a monument is 18.6m in
length. Find the height of the monument.
11) A ski tow rises 40.2m for a horizontal
distance of 120.8m. How high are you if you
have travelled 785.2m horizontally?
12) A road rises 3.8m for a horizontal
distance of 100.0m. How far have you gone
horizontally, if you have risen 32.3m