Geometry
Geometry Part II
Similar Triangles
By
Dick Gill, Julia Arnold and
Marcia Tharp
for
Elementary Algebra Math 03 online
SIMILAR TRIANGLES
Similar triangles are triangles that have the same shape but
different sizes. Corresponding angles of similar triangles are
equal and corresponding sides of similar triangles are
proportional. The nested triangles below are similar triangles.
Consider the similar
triangles to the lower
right. Since the
corresponding angles
are equal A = D,
B = E, and C = F.
A Also, the
corresponding sides
are proportional.
This means that if
BC is twice as big as
D EF, then AB has to
be twice as big as DE
and AC has to be
twice as big as DF.
B C E F
Sometimes naming angles can be confusing. In the triangle
below, you can identify angle A without any problem, but there
are actually three angles at the point D so any reference to angle
D would be confusing.
A
For this reason, we frequently name angles
with three points that trace out the path of
the angle. Angle A for example could also be
named angle CAD or angle DAC. Angle C
could also be named angle ACD.
B
D C
See if you can match the names of the angles with the numbered
angles in the sketch.
Angle ADB is angle 2
A Angle ABD is angle 4
1
Angle BAD is angle 1
Angle CBD is angle 5
Angle DBC is also angle 5
Angle ACD is angle 6
4 B
Angle BCD is also angle 6
5
2 Angle ADC is the right angle formed
D 3 6 C by combining angles 2 and 3.
Suppose that AD is perpendicular to DC and that DB is
perpendicular to AC. Remember that perpendicular lines form
right angles. Suppose also that angle ACD is 60o. Take a minute
to see if you can find the measure of angle A.
A
Remember that the angles of a triangle add up to
180o.
Solution:
A + C + ADC = 180o
B A + 60o + 90o = 180o
A + 150o = 180o
A = 30o
D C
Now see if you can find other angles in the sketch. So far we
have angle C = 60o and angle A = 30o. We also know that AD is
perpendicular to DC and that DB is perpendicular to AC.
Remember that there are three triangles in the
A sketch and that the angles of each triangle add up
to 180o.
Find angles ADC, ABD, and DBC
Solution:
Angle ADC = angle ABD = angle DBC =
90o because of the perpendicular lines.
Find angle ADB. Solution:
B
A + ADB + DBA = 180o
30o + ADB + 90o = 180o
D C
ADB + 120o = 180o so ADB = 60o
And now a True-False Question:
All three triangles in the sketch that we have been working
with are similar triangles. True or False?
A Spend some time on this before you click.
The question is really whether or not the
angles of all three triangles match up.
B
D C
And now a True-False Question:
All three triangles in the sketch that we have been working
with are similar triangles. True or False?
A Its true! It might help to redraw the smaller triangles.
Watch how the angles match up.
A
D
B
D C B C B D
ADC, DBC and ABD are all right angles. For each triangle,
the angle at the top is 30o. For each triangle, the angle at the
lower right is 60o. The triangles are similar since their
corresponding angles are equal. We denote the similarities:
ADC ~ DBC ~ ABD so that the first letter of
A each triangle represents the vertex at the top of
each triangle, the second letter represents the right
angle, etc. A
D
B
D C B C B D
Corresponding Sides of Similar Triangles are
Proportional: An Example
For the triangles below: ABC ~ DEF, AB = 8 cm, BC = 6 cm,
and DE = 5 cm. Find EF. Round to the nearest tenth.
Solution: AB DE 8 5
8x = 30
BC EF 6 x
x = 30/8
C
x = 3.8 cm
F
A B D E
Review: To solve an equation like
this, cross multiply.
8 5
6 x Multiply
8x = 30
X = 30/8 or 15/4
In the sketch below ABD ~ ECF, AB = 6 in, EC = 5 in, and CD
= 8 in. Find BD. Round to the nearest tenth.
AB BD
Solution:
EC CD
6 x
5 8 There are many
A 5 x 48 different ways to set up
a proportion and some
E x 9.6 in of them are correct.
The key is good
organization. For
example…
B C D
In the sketch below ABD ~ ECF, AB = 6 in, EC = 5 in, and CD
= 8 in. Find BD. Round to the nearest tenth.
This proportion is organized nicely because…
The numerator
of each fraction
The sides in the left fraction
comes from the
are in corresponding positions.
big triangle.
AB BD
A EC CD
E The denominator
of each fraction
comes from the
small triangle.
B C D
In the sketch below ABD ~ ECF, AB = 6 in, EC = 5 in, and CD
= 8 in. Find BD. Round to the nearest tenth.
AB BD
We have seen how works for this problem.
EC CD
What do you think about the following proportions?
AB EC
Good organization.
BD CD
A Bad organization:
AB CD
the numerators do
E BD EC
not correspond.
EC CD
Good organization.
AB BD
B C D
Practice Problems:
Which of the following triangles are similar?
B E
H I
60
60 60 D 83 83 F
A 60
C
O
K J
P
60
Q
R
90 30 30
S
M L
N
Practice Problems:
How do you write this similarity down?
B E
H I
60
60 60 D 83 83 F
A 60
C
O
K J
P
60
Q
R
90 30 30
S
M L
N
When writing down the triangles which are similar, you must match the
letters of equal angles. For example ABC is similar to HIJ with HIJ
written in any order because all the angles measure 60.
DEG is similar to SQR or you can write RQS since R and S are
equal. LMK is similar to NOP since angles L and N are equal, M and O
are equal and K and P are equal. E
B H I
60
60 60 D 83 83 F
A 60
C
O
K J
P
60
Q
R
90 30 30
S
M L
N
Once you choose the order for the first triangle,
the order for the second triangle is automatically
determined by the corresponding angles.
1. If triangle ABD is similar to triangle ECD and
AB = 10
BD = 20
EC = 8
What is CD?
A
E
B C D
If triangle ABD is similar to triangle ECD and
AB = 10
BD = 20
EC = 8
Note see how the corresponding angles
What is CD? also make corresponding sides!
Since ABD is similar to ECD then
side AB corresponds to side EC and
A
Since ABD is similar to ECD then side BD
E to side CD
We set up the proportion as:
10
AB EC 10 8
8 Or
BD CD
20 x
B C ? Or x D
20
10 8
20 x
Cross multiply
10x = 8(20)
10x = 160
x = 16
2. Triangle ABC is similar to Triangle GHF.
If AC = 34, BC = 8 and HF = 2 what is GF?
Complete Solution
3. How would you write down the similarity of the following two triangles?
B
16 18
C
A
16 Complete Solution
2. Triangle ABC is similar to Triangle GHF.
If AC = 34, BC = 8 and HF = 2 what is GF?
You don’t need pictures as long as you know the way the similarity is
written. AC and BC are in the same triangle and HF is in the other.
A B C similar to GHF
AC is first and third which corresponds to GF also first and third
letter in the similarity.
We begin to write AC GF
the proportion
as follows:
2. Triangle ABC is similar to Triangle GHF.
If AC = 34, BC = 8 and HF = 2 what is GF?
A B C similar to GHF
BC is second and third which corresponds to HF also second and third
letter in the similarity.
We finish writing AC GF
the proportion
as follows: BC HF
Now substitute the numbers:
34 X
8 2
2(34) = 8x 68 17
68 = 8x X 8.5 Return to Problem
8 2
3. Write the similarity of the two triangles.
B
146
16 18
A
C
18 N corresponds to A
16 P corresponds to B
M M corresponds to C
Since 180 - (16 + 18) = 146 All of the angles in these two triangles are equal.
So, the triangles are similar.
If you begin with triangle ABC then the correspondence would be triangle NPM.
If you began with triangle CAB then the correspondence would be triangle MNP.
End show
Go on to Part 3: Parallel Lines
Angles, and Triangles