Download Section 14.1

14.1 Ratio & Proportion
The student will learn about:
similar triangles.
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1
Triangle Similarity
Definition. If the corresponding angles in two
triangles are congruent, and the sides are
proportional, then the triangles are similar.
A
D
E
F
B
C
AB AC BC


DE DF EF
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AAA Similarity
Theorem. If the corresponding angles in two
triangles are congruent, then the triangles are
A
similar.
D
E
F
B
C
Since the angles are congruent we need to show
the corresponding sides are in proportion.
AB AC BC


DE DF EF
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If the corresponding angles in two triangles are
congruent, then the triangles are similar.
AB AC BC
Prove:
What
will
we
prove?

What
given?B=E, C=F
Given: is
A=D,
DE DF EF
Construction
Why?
(1) E’ so that AE’ = DE
(2) F’ so that AF’ = DF
(3) ∆AE’F’  ∆DEF
(4) AE’F =E =  B
Why?
Construction
SAS.
Why?
CPCTE & Given
Why?
(5) E’F’ ∥ BC
(6) AB/AE’ = AC /AF’
Corresponding angles
Why?
(7) AB/DE = AC /DF
Substitute
Why?
A
Why?
Prop Thm
(8) AC/DF = BC/EF is proven in the
same way.
QED
E’
F’
B
C
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AA Similarity
Theorem. If two corresponding angles in two
triangles are congruent, then the triangles are
similar.
A
D
E
F
B
C
In Euclidean geometry if you know two angles
you know the third angle.
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Theorem
If a line parallel to one side of a triangle
intersects the other two sides, then it cuts off a
similar triangle.
Don’t confuse this theorem
A
with If a line intersects two
sides of a triangle , and
D
E
cuts off segments
proportional to these two
sides, then it is parallel to B
the third side.
Proof for homework.
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C
SAS Similarity
Theorem. If the two pairs of corresponding
sides are proportional, and the included angles
are congruent, then the triangles are similar.
A
D
E
F
B
C
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If the two pairs of corresponding sides are proportional, and the
included angles are congruent, then the triangles are similar.
we ~
prove?
Given: is
AB/DE
=AC/DF, A=D What
What
given?
Prove:will
∆ABC
∆DEF
Construction
Why?
(1) AE’ = DE, AF’ = DF
(2) ∆AE’F’  ∆DEF
(3) AB/AE’ = AC/AF’
(4) E’F’∥ BC
(5) B =  AE’F’
(6) A =  A
(7) ∆ABC  ∆AE’F’
(8) ∆ABC  ∆DEF
A
QED
E’
B
Why?
SAS
Given & substitution (1)
Why?
Basic Proportion Thm
Why?
Why?
Corresponding
angles
Reflexive
Why?
AA
Why?
Substitute 2 & 7
Why?
D
F’
E
C
F
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SSS Similarity
Theorem. If the corresponding sides are
proportional, then the triangles are similar.
A
D
E
F
Proof for homework.
B
C
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Right Triangle Similarity
Theorem. The altitude to the hypotenuse
separates the triangle into two triangles which are
similar to each other and to the original triangle.
b
a
c-x
A
Proof for homework.
B
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Pythagoras Revisited
b
From the warm up:
a
c-x
B
A
short side
a h x
    a 2  cx
hypotenuse c b a
long side
b cx h
2
 
  b  c (c  x)
hypotenuse c
b
a
And of course then,
a 2 + b 2 = cx + c(c – x) = cx + c 2 – cx = c2
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Geometric Mean.
a b
If 
then biscalledgeometricmeanbetweenaandc.
b c
It is easy to show that b = √(ac)
m CD = 3.99 cm
m CE = 5.99 cm
E
4 6

6 9
CF = 9.00 cm
D
C
 36  36
F
or 6 = √(4 · 9)
Construction of the geometric mean.
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Summary.
• We learned about AAA similarity.
• We learned about SAS similarity.
• We learned about SSS similarity.
• We learned about similarity in right triangles.
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Assignment: 14.1