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Similar Triangles
Similarity Laws
Areas
Irma Crespo 2010
Matching
triangles
scale factor
ratio
proportion
similar polygons
congruent corresponding angles
and proportionate ratio of
corresponding side lengths
polygon with three sides
ratio of the lengths of the
corresponding sides of similar
polygons
compares two equivalent ratios
compares two quantities or
measurements in fraction form
congruent
corresponding
angles and
proportionate
side lengths
Similar Triangles We Measured
A
a
70°
D
d
70°
E e = 55°
B b = 55°
angle a = angle d = 70°
angle b = angle e = 55°
angle c = angle f = 55°
55° = f
55° = c C
AB = AC = BC = 1.9
DE DF EF
F
Fast Facts
triangle
angle
congruent
The sum of its interior angles is 180°.
degree
similar
Similarity Laws
• AA (Angle, Angle) Similarity
• SSS (Side, Side, Side) Similarity
• SAS (Side, Angle, Side) Similarity
AA Similarity
• If two angles of one triangle are congruent
to two angles of another triangle, then the
triangles are similar.
H
Two
corresponding
angles are
congruent.
K
J
G
L
I
Two
matching
angles are
congruent.
AA Similarity Examples
• Two angles of one triangle are congruent to
two angles of another triangle.
A
=
E
F
=
=
SIMILAR
B
C
D
AA Similarity Example
• Two angles of one triangle are congruent to
two angles of another triangle.
A
=
=
B
C
=
E
NOT SIMILAR
D
F
AA Similarity Example
• Two angles of one triangle are congruent to
two angles of another triangle.
C
=
E
=
D
A
F
B
These are right triangles.
=
SIMILAR
AA Similarity Example
• Two angles of one triangle are congruent to
two angles of another triangle.
E
=
C
D 90°
30°
F
60°
A 90°
=
=
SIMILAR
B
SSS Similarity
• If the measures of the corresponding sides
of two triangles are proportional, then the
triangles are similar.
There’s a scale factor.
A
D
E
B
F
C
AB = BC = AC
DE
EF
DF
SSS Similarity Example
• Corresponding side lengths are proportional.
A
7
4
B
AB = BC = AC
DE
EF
DF
C
10
E
20
8
F
14
D
4
8
7
= 14
=
1
2
SIMILAR
4
8
SSS Similarity Example
• Corresponding side lengths are proportional.
A
AB = AC = BC
DE
DF
EF
D
8
4
8
E
B
C
5.2
8
4
4
2.8
8
= 4
=
5.2
2.8
F
NOT SIMILAR
SAS Similarity
• If the measures of two sides of a triangle are
proportional to the measures of corresponding sides
of another triangle and the included angles are
congruent, then the triangles are similar.
D
A
AB
DE
E
F
= AC
DF
and
=
B
C
SAS Similarity
• If the measures of two sides of a triangle are
proportional to the measures of corresponding sides
of another triangle and the included angles are
congruent, then the triangles are similar.
D
A
10
65°
AB
DE
5 65° 6
12
E
F
= AC
DF
and
=
B
C
SAS Similarity Example
• Two sides of a triangle are proportional to the
measures of corresponding sides of another triangle
and the included angles are congruent.
D
A
10
AB
DE
5 65°
65°
E
6
F
= BC
EF
and
=
B
12
C
NOT SIMILAR
But are
they
included
angles?
Your Turn
• Determine whether the triangles are similar.
Justify your answer.
N
1 cm
L
4 cm
NM =
ZY
1
=
2
M
Z
ML
YX
4 or 1
8
2
2 cm
Y
8 cm
SIMILAR BY SAS
=
X
90°
=
90°
Your Turn
• Determine whether the triangles are similar.
Justify your answer.
A
B
=
C
=
D
SIMILAR BY AA
F
E
Areas of Similar Triangles
• Areas of similar triangles have a ratio that
equals the square of the scale factor.
Scale factor is 2.
Scale factor is 1/3.
The square is 4.
The square is 1/6.
Scale factor is ¼.
The square is 1/16.
Scale factor is 5.
The square is 25.
To square is to multiply the number by itself.
Finding Areas of Similar Triangles
• Triangle GHI and triangle JKL are similar with a
scale factor of 3. If the area of triangle GHI is 81
square meters, find the are of triangle GHI.
I
10 cm
7 cm
G
4 cm H
K
30 cm
21 cm
J
12 cm L
GHI
JKL
9 81
=
1 x
9x = 1*81
9x = 81
9x = 81
9
9
x = 9 cm
Find the Scale Factor of Two Areas
• The two triangles below are similar. The scale factor
is the ratio of the corresponding sides. Find the scale
factor of the two triangles. Then, find the scale factor
of their areas.
Y
10 in
6 in
E
X
21 in
G
8 in
35 in
28 in
F
Z
Find the Scale Factor of Two Areas
Find the scale factor.
Y
XYZ
10 in
6 in
X
8 in
GEF
Z
Find the scale factor of the areas.
Use A = ½ *base*height
E
Area of
21 in
35 in
6 or 2
21
7
Area of
1
2
XYZ = 2 (8*6) = 24 in
GEF = 1 (28*21) = 294 in2
2
Ratio of the areas.
G
28 in
F
24 or 4 .
294 49
Summary
• Triangles are similar if they show AA
Similarity, SSS Similarity, or SAS
Similarity.
• The areas of similar triangles are the ratio
that equals the square of the scale factor.
• To find the scale factors of two areas, just
compute for the areas of each triangle and
then, form the ratio of the areas.
Exit Slip
• Explain one of the Similarity Laws
discussed today by either giving its meaning
or making your own example or both.
• You will get an extra credit for this.
• Write your name and submit before the end
of the hour.
Practice Worksheet
Complete the practice worksheet.
Exercises MI35 Lesson 6
Work with a partner or on your own.
Submit completed worksheet for grading.
Solutions are discussed the next day.
Main Resources
• Lesson Plan Problems Math Connects: Concepts,
Skills, and Problem Solving; Teacher Edition;
Course 3, Volume 1 Columbus:McGraw-Hill, 2009.
• PowerPoint created by Irma Crespo.
University of Michigan-Dearborn, School of
Education. Winter 2010.