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McDougal Geometry chapter 6 notes
6.1 Ratios, proportions, and the geometric mean
We will use first parts of TK#50 in 6.1 and the rest in 6.2.
TK#50: Properties of proportions
The geometric mean of 2 positive numbers a and b is the positive number x such that
a x
 .
x b
1.
So x2=ab and
x  ab.
Cross product property. In a proportion, the product of the extremes equals the
product of the means. If
2.
where b≠0 and d≠0, then ad=bc.
Reciprocal property. If 2 ratios are equal, then their reciprocals are also equal.
If
3.
a c

b d
a c

b d
, then
b d
 .
a c
If we swap the means of a proportion, then we form another true proportion. If
a c

b d
, then
a b
 .
c d
In a proportion, if we add the value of each ratio’s denominator to its numerator, then we
form another true proportion. If
a c

b d
, then
ab cd

.
b
d
GP 2-4,6,7,11
Hwk: p360#7-65 odds (skip 55).
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6.2 Use proportions to solve geometry problems, pg. 367 #3-17 odds, 18, 23-29
Example 1 and GP1.
GP2.
Example 3 and GP3.
GP4.
GP5.
(If time) Prove: If
a c
ab cd
 , then

.
b d
b
d
2
6.3 Use similar polygons
2 polygons are similar (~) if corresponding angles are congruent and corresponding
sides are proportional. If ABCD~EFGH, then corresponding angles are congruent:
A  E , B  F , C  G, D  H and corresponding sides are
AB BC CD DA



. In writing the similarity statements, order is
proportional:
EF FG GH HE
crucial!
B
F
A
D
E
G
H
C
TK#51: Similar polygons
Any corresponding 1-dimensional part (length, perimeter, radius, etc) of similar polygons
is equal to the scale factor.
Guided practice 1,4-6.
6.3 one day pg. 376#5-27 odds, 31, 36 or Section 6.3 day 1 pg. 376 #1-11, 40-45
Section 6.3 day 2 pg. 376 #12-26, 31-32
__________ Quiz Review pg. 380 #1-7
__________Quiz 6.1-6.3. Hwk: Activity on the top of pg 381: Angles and similar
triangles (need protractor and ruler). Steps 1-4. Draw conclusions 1,2.
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6.4 Prove triangles similar by AA, pg 384#3-25 odds, 32-36E
Turn in Activity 6.4.
TK#52: AA similarity postulate: If 2 angles in one triangle are congruent to 2 angles in
another triangle, then the 2 triangles are similar. Since J  X , K  Y , then
∆JKL~∆XYZ by AA∆~Post.
K
Y
L
Z
X
J
Guided practice 2, 4.
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6.5 Prove triangles similar by SSS and SAS
TK#53: SSS Similarity Thm: If all 3 corresponding sides of two triangles are
proportional, then the triangles are similar. If
AB BC CD


, then ∆ABC~∆RST
RS ST TR
by SSS∆~ Thm.
A
R
C
T
S
B
TK#54: SAS Similarity Thm: If 2 sides of one triangle are proportional to the
corresponding 2 sides of another triangle and the included angles are congruent, then the
two triangles are similar. If
JK KL

and K  Y , then ∆JKL~∆XYZ by
XY YZ
SAS∆~ Thm.
K
Y
L
Z
X
J
Find x to make ∆JKL~∆MYZ.
Example
K
3(x-2)
Y
12
L
21
Z
x+6
20
30
M
J
Guided practice 1-3.
5
If time: prove guided practice 4 using 2 different methods.
6.5 one day: pg 391 #3-23 odds, 29-35 odds or 6.5 day 1 pg 391 #3-23 odds, 39-44
(bring compass and straightedge tomorrow)
6.5 day 2 pg 393 #18-24E, 28-36E (bring compass and straightedge tomorrow)
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6.6 Use proportionality thm, pg 400 #3, 9-18
TK#55: Angle Bisector Proportionality Thm: If a ray bisects an angle of a triangle, then
it divides the opposite side into the same proportion as the 2 sides that form the angle.
AD AC

DB BC
A
D
C
B
Example 4.
Dilation Activity 6.7 on pg 408 Steps 1-4. Draw conclusions 1,2. (Need graph paper,
compass, and ruler.) Hwk: Ch 6 review pg 418 #1-18. Print out Ch 7 notes.
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6.7 Perform similarity transformations (Need graph paper). Pg 412 #3-21 odds (skip 17)
Turn in: Activity 6.7, ch 6 review, and printed ch 7 notes.
TK#56 Similarity transformation:
With dilation: If 0<k<1, reduction. If k>1, enlargement.
Guided practice 2, 3.
_______ Ch 6 Group Quiz. Collect TK#1-56.
_______ Correct ch 6 gp quiz.
_______ Ch 6 individual Test
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