Download 8.2 Warm Up x 1. 2.

8.2 Warm Up
Find the value of x.
1.
2.
3.
February 18, 2016
4.
Geometry 8.2 Proving Triangle Similarity by AA
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8.2 Warmup
Find the indicated measure in
1. ML
2. MJ
3. JN
4. MK
5. m∠MJK
6. m∠LMJ
7. m∠MKL
8. m∠LJM
February 18, 2016
JKLM.
Geometry 8.2 Proving Triangle Similarity by AA
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Geometry
8.2 Proving Triangle Similarity by AA
8.2 Essential Question
What can you conclude about two triangles
when you know that two pairs of corresponding
angles are congruent?
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Geometry 8.2 Proving Triangle Similarity by AA
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Goals



Identify similar triangles.
Prove triangles similar.
Solve problems using similar triangles.
~
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Geometry 8.2 Proving Triangle Similarity by AA
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Consider these two triangles…
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Geometry 8.2 Proving Triangle Similarity by AA
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These two angles are
congruent…
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Geometry 8.2 Proving Triangle Similarity by AA
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These two angles are
congruent…
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Geometry 8.2 Proving Triangle Similarity by AA
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And these two angles are
congruent…
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Geometry 8.2 Proving Triangle Similarity by AA
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Two triangles, three congruent
angles.
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Geometry 8.2 Proving Triangle Similarity by AA
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These triangles are similar.
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Do we really need to show all three
angles congruent?
NO!
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What must the last angle in
each triangle be?
?
?
70
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70
30
30
Geometry 8.2 Proving Triangle Similarity by AA
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The sum of the angles in each
triangle must be 180.
80
80
70
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70
30
30
Geometry 8.2 Proving Triangle Similarity by AA
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Third Angles Thm
We only need to know two angles
are congruent. The remaining
angles must also be congruent.
70
70
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30
30
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AA Postulate
(Angle-Angle)
If two angles of one triangle
are congruent to two angles
of another triangle, then the
triangles are similar.
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Geometry 8.2 Proving Triangle Similarity by AA
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M
Example 1
Is BAT ~ MAN?
A
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B
T
Geometry 8.2 Proving Triangle Similarity by AA
N
17
M
Example 1 Solution
Is BAT ~ MAN?
A
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B
T
Geometry 8.2 Proving Triangle Similarity by AA
N
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Example 1 Solution
M
Is BAT ~ MAN?
A
N
B
Yes.
Why?
A
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T
AA Postulate
Geometry 8.2 Proving Triangle Similarity by AA
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M
Example 1 Solution
BAT ~ MAN
A
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B
T
Geometry 8.2 Proving Triangle Similarity by AA
N
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Abbreviation
Instead of writing:
 AA Postulate
 or AA Similarity Postulate
We will abbreviate this as:

AA~
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Your Turn 1
Are these triangles
similar?
F
T
If so, why?
If yes, write a
statement of similarity.
A
N
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Geometry 8.2 Proving Triangle Similarity by AA
M
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Your Turn 1 Solution
FAT  MAN
F
T
Why?
Vertical angles.
A
N
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Geometry 8.2 Proving Triangle Similarity by AA
M
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Your Turn 1 Solution
T  N
F
T
Why?
Alternate Interior
Angles
A
N
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Geometry 8.2 Proving Triangle Similarity by AA
M
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Your Turn 1 Solution
Are the triangles
similar?
F
T
Yes.
Why?
AA~
A
(Notice, F  M, but
we only need two
congruent angles.)
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Geometry 8.2 Proving Triangle Similarity by AA
N
M
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Your Turn 1 Solution
Statement of Similarity:
F
T
FAT ~MAN
Be sure to match
up corresponding
vertices correctly!
A
N
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Geometry 8.2 Proving Triangle Similarity by AA
M
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Example 2
a. Are these triangles similar? If yes, write
a similarity statement.
Y
85
65
X
Yes, they are
similar.
30
30
Z
K
J
85
65
H
XYZ ~ HJK
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Example 2
b. List all the pairs of congruent angles
and write the statement of proportionality.
Y
X  H
85
J
Y  J
85
Z  K
65
X
30
30
Z
K
65
H
𝑋𝑌
𝑋𝑍
𝑌𝑍
=
=
𝐻𝐽
𝐻𝐾
𝐽𝐾
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Your Turn 2
a. Are these triangles similar? If yes, write
a similarity statement.
N
Yes, they are similar by AA~.
NBA ~ NFL
B
F
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A
L
Geometry 8.2 Proving Triangle Similarity by AA
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Your Turn 2
b. List all the pairs of congruent angles
and write the statement of proportionality.
N  N
N
NBA  NFL
B
F
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NAB  NLF
A
𝑁𝐵 𝑁𝐴
𝐵𝐴
=
=
𝑁𝐹
𝐹𝐿
𝑁𝐿
L
Geometry 8.2 Proving Triangle Similarity by AA
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Example 3
Show ∆𝐴𝐵𝐶~∆𝐴𝐷𝐸. Find AD.
Since 𝐵𝐶 ∥ 𝐷𝐸,
∠𝐴𝐵𝐶 ≅ ∠𝐴𝐷𝐸.
∠𝐴 ≅ ∠𝐴, by the
reflexive property.
So, ∆𝐴𝐵𝐶~∆𝐴𝐷𝐸
by AA~
A
16
B
C
10
D
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Geometry 8.2 Proving Triangle Similarity by AA
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Example 3
𝐴𝐵 𝐴𝐶
=
𝐴𝐷 𝐴𝐸
16 20
=
𝐴𝐷 30
A
24
20𝐴𝐷 = 480
𝐴𝐷 = 24
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16
B
?
20
30
C
10
D
Geometry 8.2 Proving Triangle Similarity by AA
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Your Turn 3
Solve for x.
x
24
20
45
20 24

45 x
20 x  1080
x  54
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Example 4
The triangles are similar. Find the value of
16
the variable.
24
x+4
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10
x  4 24

16
10
10 x  40  384
10 x  344
x  34.4
Geometry 8.2 Proving Triangle Similarity by AA
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Example 5
Jimmy, who is 36 inches tall, casts a
shadow that is 48 inches long.
A nearby windmill has a shadow
that is 64 feet long. How tall is the
windmill?
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Geometry 8.2 Proving Triangle Similarity by AA
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Example 5 Solution
Jimmy, who is 36 inches tall, casts a
shadow that is 48 inches long.
A nearby windmill has a shadow
that is 64 feet long. How tall is the
windmill?
First, get
uniform units!
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Example 5 Solution
Jimmy, who is 3 feet tall, casts a
shadow that is 4 feet long.
A nearby windmill has a shadow
that is 64 feet long. How tall is the
windmill?
x ft
64 ft
3 ft
4 ft
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Now label the
drawing.
Geometry 8.2 Proving Triangle Similarity by AA
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Example 5 Solution
3 x

4 64
x
64
3
4
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Set up the
proportion.
Geometry 8.2 Proving Triangle Similarity by AA
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Example 5 Solution
3 x

4 64
4 x  192
x
64
x  48
3
Solve it.
4
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Example 5 Solution
The windmill is 48 feet tall.
48
64
3
Answer it.
4
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Geometry 8.2 Proving Triangle Similarity by AA
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Your Turn 5
The Washington Monument is 169.3 m tall, and the
base is 16.8 m on each side.
For a class project, you are going to make a scale
model that is going to be 2 m tall. What will the length
of each side of the base be?
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Your Turn 5
169.3 2

16.8 x
169.3 x  33.6
169.3
x  0.1984...
x  0.2 m
2
?
16.8
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Generalization






If any two polygons are similar, the sides are
proportional.
This also can be extended to:
Altitudes
Medians
Diagonals
any corresponding lengths.
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Example 6
MAD ~ CAP
D
M
Find x.
20
24
A
x
P
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C
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Example 6 Solution
Since MAD ~ CAP,
sides and altitudes are
proportional:
24 20

10 x
24 x  200
x 8
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1
3
D
M
20
24
A
x
P
Geometry 8.2 Proving Triangle Similarity by AA
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C
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Your Turn 6
The figures are similar. Find the length of
the diagonal of the larger one.
3
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8
8
Geometry 8.2 Proving Triangle Similarity by AA
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Your Turn 6 Solution
3
8
sides

8 d
3d  64
d  21
3
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8
diagonals
1
3
~
8
Geometry 8.2 Proving Triangle Similarity by AA
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Summary



Two triangles are similar if two angles of one
triangle are congruent to two angles of the
other triangle. (AA~ postulate)
If triangles are similar, then their sides are
proportional.
Also, medians, altitudes, and other
corresponding lengths are proportional.
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Homework
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