Transcript Chapter 7: Proportions and Similarity

Chapter 7:
Proportions and Similarity
7.1- Proportions
 Make a Frayer foldable
7.1 Ratio
and
Proportion
Ratio
 A comparison of two quantities using division
 3 ways to write a ratio:



a to b
a
b
a:b
Proportion
 An equation stating that
two ratios are equal

Example: a  c
b d
 Cross products: means
and extremes

Example:
a c

b d
ad = bc
a and d = extremes
b and c = means
Your Turn: solve these examples
Ex:
3 21

x 6
Ex:
x2 4

2
5
3 * 6 = x * 21
5(x – 2) = 2 * 4
18 = 21x
5x – 10 = 8
x = 18/21
5x = 18
x = 6/7
x = 18/5
x = 3 3/5
Your Turn: solve this example
 The ratios of the measures of three angles of
a triangle are 5:7:8. Find the angle measures.
5x + 7x + 8x = 180
20x = 180
x=9
45, 63, 72
7.2 : Similar Polygons
 Similar polygons have:
 Congruent corresponding angles
 Proportional corresponding sides
A
Polygon ABCDE ~ Polygon LMNOP
B
L
E
M
C
D
P
N
Ex:
AB CD

LM NO
O
 Scale factor: the ratio of corresponding sides
7.3: Similar Triangles
 Similar triangles have
congruent
corresponding angles
and proportional
corresponding sides
Z
Y
A
C
X
B
angle A  angle X
ABC ~
XYZ
angle B  angle Y
angle C  angle Z
AB AC BC


XY XZ YZ
7.3: Similar Triangles
 Triangles are similar if you show:



Any 2 pairs of corresponding sides are
proportional and the included angles are
congruent (SAS Similarity)
All 3 pairs of corresponding sides are
proportional (SSS Similarity)
Any 2 pairs of corresponding angles are
congruent (AA Similarity)
7.4 : Parallel Lines and Proportional
Parts
 If a line is parallel to
one side of a triangle
and intersects the other
two sides of the
triangle, then it
separates those sides
into proportional parts.
A
Y
*If XY ll CB, then
AY AX

YC
XB
C
X
B
7.4 : Parallel Lines and Proportional
Parts
 Triangle Midsegment
Theorem

A midsegment of a
triangle is parallel to
one side of a triangle,
and its length is half of
the side that it is
parallel to
*If E and B are the midpoints
of AD and AC respectively,
1
then EB = 2 DC
A
E
D
B
C
7.4 : Parallel Lines and Proportional
Parts
 If 3 or more lines are
parallel and intersect
two transversals, then
they cut the
transversals into
proportional parts
A
B
C
D
E
F
AB DE

BC EF
AC BC

DF EF
AC DF

BC EF
7.4 : Parallel Lines and Proportional
Parts
 If 3 or more parallel
lines cut off congruent
segments on one
transversal, then they
cut off congruent
segments on every
transversal
If
AB  BC , then DE  EF
A
B
C
D
E
F
7.5 : Parts of Similar Triangles
 If two triangles are
X
similar, then the
perimeters are
proportional to the
measures of
corresponding sides
A
B
C
Y
perimeterABC AB BC AC



perimeterXYZ XY YZ
XZ
Z
7.5 : Parts of Similar Triangles
 If 2 triangles are similar,
 If 2 triangles are similar,
the measures of the
corresponding altitudes
are proportional to the
corresponding sides
the measures of the
corresponding angle
bisectors are
proportional to the
corresponding sides
X
A
S
M
B
C
D
Y
W
AD AC BA BC



XW XZ YX YZ
Z
L
R
O
N
U
MO MN LM LN



SU
ST
RS
RT
T
7.5 : Parts of Similar Triangles
 An angle bisector in a
 If 2 triangles are similar,
triangle cuts the opposite
side into segments that are
proportional to the other
E
sides
then the measures of
the corresponding
medians are
proportional to the
corresponding sides.
A
BC AB

CD AD
G
T
B
J
H
D
C
I
F
U
GI
GH
GJ
HJ



TV
UT
TW
UW
V
W
G
FG EF

GH EH
H