3.9.3 Similar Triangle Properties
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Similar Triangle Properties The student is able to (I can): • Use properties of similar triangles to find segment lengths. • Apply proportionality and triangle angle bisector theorems. • Apply triangle angle bisector theorems • Use triangle similarity to solve problems.
Transcript of 3.9.3 Similar Triangle Properties
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Similar Triangle Properties
The student is able to (I can):
Use properties of similar triangles to find segment lengths.
Apply proportionality and triangle angle bisector theorems.
Apply triangle angle bisector theorems
Use triangle similarity to solve problems.
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Triangle Proportionality Theorem
If a line parallel to a side of a triangle intersects the other two sides then it divides those sides proportionally.
S
P
A
C
E
>
>
PC SE
AP AC
PS CE=
Note: This ratio is not the same as the ratio between the third sides!
AP PC
PS SE
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Triangle Proportionality Theorem Converse
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
S
P
A
C
E
>
>
PC SE
AP AC
PS CE=
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Two Transversal Proportionality
If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.
G
O
D
T
A
C>
>
>
CA DO
AT OG=
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Examples Find PE
10x = (4)(14)
10x = 56
S
C
O
P
E
10101010 14141414
4444
10 14
4 x=
xxxx
28 3x 5 5.6
5 5= = =
>
>
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Example Verify that
(15)(8) = (10)(12)?
120 = 120 Therefore,
H
O
RSE
HE OS
15
10
12 8
=15 10
?12 8
HE OS
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Example Solve for x.
6x = (10)(9)
6x = 90
x = 15
>
>
>
x
96
10
10 x
6 9=
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Triangle Angle Bisector Theorem
An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.
=CD CA
DB AB
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Example: Solve for x.
=AD AB
DC BC
=
=
= =
3.5 5
x 125x 42
42x 8.4
5
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Angle-Angle Similarity (AA~)
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
M P
A O
Therefore, MAC ~ POD by AA~
M
A C
P
O
D
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Side-Side-Side Similarity (SSS~)
If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar.
W H
Y
N
O
T
= =WH HY WY
NO OT NT
Therefore, WHY ~ NOT by SSS~
1230
18
16
40
24
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Side-Angle-Side Similarity (SAS~)
If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar.
E
T X
U
L V
=LU LV
TE TX L T
Therefore, LUV ~ TEX by SAS~
4
5
2
2.5
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Example Explain why the triangles are similar and write a similarity statement.
90 56 = 34
Therefore mV = mX, thus V X.
Since mU = mE = 90, U E
Therefore, LUV ~ TEX by AA~
56
34L
U V
T
E
X
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Example Verify that SAT ~ ORT
A
ST
R
O
12
15
20
16
ATS RTO (Vertical angles )
12 15?
16 20=
240 = 240Therefore, SAT ~ ORT by SAS~
