Download Similar Triangles - Peoria Public Schools

similar_triangles.notebook
March 25, 2013
Similar Triangles
1) The student will learn proportional relationships
2) The student will be able to set up proportional relationships between the sides of triangles
3) The student will be able to use and describe AA Similarity Postulate and SSS or SAS Similarity Theorem Sue Blakely
SMART Exemplary Educator
El Paso, TX
Lesson objectives
Teachers' notes
1
similar_triangles.notebook
March 25, 2013
Subject: Geometry
Topic: Similar Triangles
Grade(s): 10
Prior knowledge: proportional relationships, fractions, triangles, cross­product multiplication, congruent angles
Lesson notes:
Page 8: Let students pull apart triangles and line them up to better see the similarity. Use math tools to compare the congruent angle measurements. (If you don't have math tools, press link and copy of angles will come up.) Page 11: Students fill the appropriate lines of the black triangles with corresponding colors and label the measures. Students drag the letters to the appropriate places on the triangles and angles (cloned letters)
Lesson objectives
Teachers' notes
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similar_triangles.notebook
March 25, 2013
Ratios ­ Means & Extremes
extreme
=
mean
extreme a
mean b
=
mean
extreme
c mean
d extreme
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similar_triangles.notebook
March 25, 2013
Ratios ­ Means & Extremes
15
20
=
30
40
Name the means:
Name the extremes:
4
similar_triangles.notebook
March 25, 2013
Cross Products Property
In a proportion, the product of the extremes equals the product of the means. c
a
If = where b ≠ 0 and d ≠ 0, then ad = bc
d
b
2
3
=
4
6
3 x 4
=
2 x 6
= 12
5
similar_triangles.notebook
March 25, 2013
Geometric Mean
The geometric mean of two positive numbers a and b is the positive number x that satisfies x
a
2
b So, x = ab and x = √ab x = . 5 15
Given = , find the geometric mean.
9
3
x = √ab x = √5x9 x = √45
x = 3√5
6
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Geometric Mean
What is the geometric mean: 2 and 18
4 and 25
2 and 25
6 and 20
7
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What makes these two sets of polygons have similar ratios:
Explore......
link to angle measures
8
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March 25, 2013
Similar polygons have equal corresponding angle measures
71°
71°
64°
45.1°
64°
45.1°
75.2°
122°
75.2°
122°
59.9°
249.7°
33.2°
59.9°
249.7°
33.2°
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similar_triangles.notebook
March 25, 2013
ΔRST ΔXYZ
T
R S T X
Y
Z
20
25
R
18
15
Y
30
S
X
12
Z
Similar triangles have congruent corresponding angles.
If the corresponding angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are similar. 10
similar_triangles.notebook
March 25, 2013
ΔZTX ΔRYS
Z T X R
Y
S
Arrange the color lines to the appropriate similarity and give the angle congruence.
15 12
=
20 16
R Z Y X ST
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similar_triangles.notebook
March 25, 2013
ΔRST ΔXYZ
T
R S T X
Y
Z
20
25
R
18
15
Y
30
S
X
12
Z
All similar triangles have the same ratio of corresponding sides. (SCALE FACTOR)
The Scale Factor is a ratio of one corresponding side of one triangle to the other corresponding side of the second triangle. This is written as 3:5 or 5:3 depending how the triangles are compared.
15 12 18
25 20 30
12
similar_triangles.notebook
March 25, 2013
ΔRST ΔXYZ
Set up a proportion and solve for x. Give the scale factor
T
x
25
30
=
15
Y
R
x
18
30
X
S
12
Z
12
Set up corresponding proportion
18
Solve for your variable
Scale factor 15:25 reduced is 3:5
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similar_triangles.notebook
March 25, 2013
ΔABC ΔDEF
Set up a proportion and solve for x and y.
Give the scale factor.
F
B
12
C
x
24
10
8
A
D
y
E
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Angle­Angle (AA) Similarity Postulate
If two angles of one triangle
are congruent to two angles
of another triangle, then
the two triangles are A
similar.
B
C
E
F
D
ΔABC ΔDEF
16
similar_triangles.notebook
March 25, 2013
Angle­Angle (AA) Similarity Postulate
A
Show
ΔABE ΔACD
E
52o
B
(Hint: Redraw the triangles separately)
D
52o
C
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similar_triangles.notebook
March 25, 2013
Side­Side­Side (SSS) Similarity Theorem
If the corresponding side lengths of two triangles are
proportional, then the two triangles are similar.
B
C
A
AB BC CA
= =
DE EF FD
ΔABC ΔDEF
8
12
16
E
9
D
6
F
12
4 4 4
= =
3 3 3
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Side­Side­Side (SSS) Similarity Theorem
Using the SSS Similarity Theorem, explain which set of triangles are similar. Write a similarity statement and give the scale factor.
Y
5
10
X
Z
20
F
25 D
8
16 M
2
4
L
4
N
10
E
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similar_triangles.notebook
March 25, 2013
Side­Angle­Side (SAS) Similarity Theorem
B
If an angle of one triangle is congruent to an angle of a second triangle and the
lengths of the sides including these 16
angles are proportional,
then the triangles are similar.
30o
AB AC
=
DE DF
ΔABC ΔDEF
A
E
C
12 12
30o
D
9
A D
F
4 4
=
3 3
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similar_triangles.notebook
March 25, 2013
Side­Angle­Side (SAS) Similarity Theorem
Explain using the SAS Similarity Theorem why these two triangles are similar.
D
20
30
15
B
C
5
10
E
10
A
ΔABC ΔDEC
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similar_triangles.notebook
March 25, 2013
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