Geometry

Geometry Part IISimilar Triangles

ByDick Gill, Julia Arnold and

Marcia Tharpfor

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.

A

B C E

D

F

Consider the similar triangles to the lower right. Since the corresponding angles are equal A = D, B = E, and C = F. Also, the corresponding sides are proportional. This means that if BC is twice as big as EF, then AB has to be twice as big as DE and AC has to be twice as big as DF.

A

B

CD

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.

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.

A

B

CD

See if you can match the names of the angles with the numbered angles in the sketch.

1

23

45

6

Angle ADB is angle 2

Angle ABD is angle 4

Angle BAD is angle 1

Angle CBD is angle 5

Angle DBC is also angle 5

Angle ACD is angle 6

Angle BCD is also angle 6

Angle ADC is the right angle formed by combining angles 2 and 3.

A

B

CD

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.

Remember that the angles of a triangle add up to 180o.

Solution:

A + C + ADC = 180o

A + 60o + 90o = 180o

A + 150o = 180o

A = 30o

A

B

CD

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 sketch and that the angles of each triangle add up to 180o.

Find angles ADC, ABD, and DBC Solution:

Find angle ADB. Solution:

A + ADB + DBA = 180o

30o + ADB + 90o = 180o

ADB + 120o = 180o so ADB = 60o

Angle ADC = angle ABD = angle DBC = 90o because of the perpendicular lines.

A

B

CD

And now a True-False Question:

All three triangles in the sketch that we have been working with are similar triangles. True or False?

Spend some time on this before you click. The question is really whether or not the angles of all three triangles match up.

A

B

CD

And now a True-False Question:

All three triangles in the sketch that we have been working with are similar triangles. True or False?

Its true! It might help to redraw the smaller triangles. Watch how the angles match up.

B C

D

A

B D

A

B

CD B C

D

A

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 each triangle represents the vertex at the top of each triangle, the second letter represents the right angle, etc.

Corresponding Sides of Similar Triangles are Proportional: An Example

A B

C

D E

F

For the triangles below: ABC ~ DEF, AB = 8 cm, BC = 6 cm, and DE = 5 cm. Find EF. Round to the nearest tenth.

Solution:EF

DE

BC

AB

x

5

6

8 8x = 30

x = 30/8

x = 3.8 cm

x

5

6

8

Review: To solve an equation likethis, cross multiply.

Multiply

8x = 30

X = 30/8 or 15/4

A

B C D

E

In the sketch below ABD ~ ECF, AB = 6 in, EC = 5 in, and CD = 8 in. Find BD. Round to the nearest tenth.

Solution:

inx

x

xCD

BD

EC

AB

6.9

48585

6

There are many different ways to set up a proportion and some of them are correct. The key is good organization. For example…

A

B C D

E

In the sketch below ABD ~ ECF, AB = 6 in, EC = 5 in, and CD = 8 in. Find BD. Round to the nearest tenth.

CD

BD

EC

AB

This proportion is organized nicely because…

The numerator of each fraction comes from the big triangle.

The denominator of each fraction comes from the small triangle.

The sides in the left fraction are in corresponding positions.

The sides in the right fraction

are in corresponding positions.

A

B C D

E

In the sketch below ABD ~ ECF, AB = 6 in, EC = 5 in, and CD = 8 in. Find BD. Round to the nearest tenth.

CD

BD

EC

ABWe have seen how works for this problem.

What do you think about the following proportions?

CD

EC

BD

AB Good organization.

BD

CD

AB

EC Good organization.

EC

CD

BD

AB

Bad organization: the numerators do not correspond.

Practice Problems:

Which of the following triangles are similar?

60 60

60

60AD

H

K

N

F

EB

C

J

I

M L

O

P

QR

S

83 83

83

14

90 30

60

30

Practice Problems:

How do you write this similarity down?

60 60

60

60AD

H

K

N

F

EB

C

J

I

M L

O

P

QR

S

83 83

83

14

90 30

60

30

60 60

60

60AD

H

K

N

F

E

B

C

J

I

M L

O

P

QR

S

83 83

83

14

90 30

60

30

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.

Once you choose the order for the first triangle, the order for the second triangle is automatically determined by the corresponding angles.

A

B C D

E

1. If triangle ABD is similar to triangle ECD and AB = 10BD = 20EC = 8What is CD?

A

B C D

E

If triangle ABD is similar to triangle ECD and AB = 10BD = 20EC = 8What is CD?

108

20

? Or x

Since ABD is similar to ECD thenside AB corresponds to side EC andSince ABD is similar to ECD then side BD to side CD

CD

EC

BD

AB

We set up the proportion as:

Or

x

8

20

10

Note see how the corresponding anglesalso make corresponding sides!

x

8

20

10

Cross multiply10x = 8(20)10x = 160x = 16

2. Triangle ABC is similar to Triangle GHF. If AC = 34, BC = 8 and HF = 2 what is GF?

3. How would you write down the similarity of the following two triangles?

A

B

C

M N

P

16 18

16

146

Complete Solution

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 iswritten. AC and BC are in the same triangle and HF is in the other.

A B C similar to G H F

AC is first and third which corresponds to GF also first and thirdletter in the similarity.

GFAC

We begin to writethe 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 G H F

BC is second and third which corresponds to HF also second and thirdletter in the similarity.

HF

GF

BC

AC

We finish writingthe proportion as follows:

Now substitute the numbers:

28

34 X

2(34) = 8x68 = 8x 5.8

2

17

8

68X Return to Problem

A

B

C

N

P

16 18

16

146

M

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.

18

146

N corresponds to AP corresponds to BM corresponds to C

3. Write the similarity of the two triangles.

End show

Go on to Part 3: Parallel LinesAngles, and Triangles