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Similar Triangles
Chapter 7-3
TARGETS
• Identify similar triangles.
• Use similar triangles to solve problems.
Content Standards
G-SRT.2 and G-SRT.3 Understand similarity in terms of similarity
transformations
G-SRT.4 and G-SRT.5 Prove theorems using similarity
Mathematical Practices
1 Make sense of problems and persevere in solving them
2 Reason abstractly and quantitatively.
4 Model with mathematics
6 Attend to precision.
Triangle Similarity is:
Lesson 3 TH2
Writing Proportionality Statements
Given BTW ~ ETC
• Write the Statement of Proportionality
• Find mTEC
E
• Find TE and BE
20
EC TE TC


BW TB TW
mTEC = mTBW = 79o
EC TE
3
x

 
BW TB 12 20
T
3
34o
C
79o
TE  5 B
12
EB  20  5  15
W
AA  Similarity Theorem
• If two angles of one triangle are congruent
to two angles of another triangle, then the
two triangles are similar.
If K  Y and J  X,
K
then  JKL ~  XYZ.
Y
J
L
X
Z
Example
• Are these two triangles similar? Why?
N
M
P
Q
R
S
T
SSS  Similarity Theorem
• If the corresponding sides of two triangles
are proportional, then the two triangles are
similar.
AB BC CA
if


PQ QR RP
B
then ABC ~ PQR
Q
A
C
P
R
Which of the following three triangles are similar?
B
9
6
A
H
E
4
D
6
8
14
6
12
J
10
C
F
ABC and FDE?
AC 12
Longest Sides


FE
8
BC
6
Shortest Sides
 
DE
4
AB
9
Remaining Sides
 
FD 6
3
2
3
2
3
2
G
ABC~ FDE
SSS ~ Thm
Scale Factor =
3:2
Which of the following three triangles are similar?
B
9
6
A
H
E
4
D
6
8
14
6
12
J
10
C
F
ABC and GHJ
AC 12 6
Longest Sides


GJ 14 7
BC
6
Shortest Sides
 1
HJ
6
AB
9
Remaining Sides

GH
10
G
ABC is not
similar to DEF
SAS  Similarity Theorem
• If one angle of one triangle is congruent to an
angle of a second triangle and the lengths of
the sides including these angles are
proportional, then the triangles are similar.
K
if J  X and
JK
JL

XY XZ
then ΔJKL ~ ΔXYZ
Y
J
L
X
Z
ass
Pantograph
Prove RTS ~ PSQ
S  S (reflexive prop.)
S
SP SQ

SR ST
4
5

 4( 20)  5(16)
16 20
80  80
4
P
12
5
Q
15
SPQ  SRT
SAS  ~ Thm.
R
T
Are the two triangles similar?
N
P
NQ 12 4
 
QT
9 3
PQ 15 3
 
RQ 10 2
15
12
Q
10
R
NQP  TQR
9
T
Not
Similar
How far is it
across the river?
2
yds
x yards
5 yds
2
5

42 x
2x = 210
x = 105 yds
42
yds
Are Triangles Similar?
In the figure,
, and ABC and DCB are right
angles. Determine which triangles in the figure are
similar.
Are Triangles Similar?
by the Alternate Interior
Angles Theorem.
Vertical angles are congruent,
Answer: Therefore, by the AA Similarity Theorem,
ΔABE ~ ΔCDE.
In the figure, OW = 7, BW = 9,
WT = 17.5, and WI = 22.5.
Determine which triangles in
the figure are similar.
A. ΔOBW ~ ΔITW
B. ΔOBW ~ ΔWIT
C. ΔBOW ~ ΔTIW
D. ΔBOW ~ ΔITW
Parts of Similar Triangles
ALGEBRA Given
, RS = 4, RQ = x + 3,
QT = 2x + 10, UT = 10, find RQ and QT.
Parts of Similar Triangles
Since
because they are alternate interior angles. By AA
Similarity, ΔRSQ ~ ΔTUQ. Using the definition of similar
polygons,
Substitution
Cross products
Parts of Similar Triangles
Distributive Property
Subtract 8x and 30 from each
side.
Divide each side by 2.
Now find RQ and QT.
Answer: RQ = 8; QT = 20
A. ALGEBRA Given
AB = 38.5, DE = 11,
AC = 3x + 8, and CE = x + 2, find AC.
A. 2
B. 4
C. 12
D. 14
Lesson 3 CYP2
B. ALGEBRA Given
AB = 38.5, DE = 11,
AC = 3x + 8, and CE = x + 2, find CE.
A. 2
B. 4
C. 12
D. 14
Indirect Measurement
INDIRECT MEASUREMENT
Josh wanted to measure the
height of the Sears Tower in
Chicago. He used a 12-foot
light pole and measured its
shadow at 1 P.M. The length
of the shadow was 2 feet.
Then he measured the
length of the Sears Tower’s
shadow and it was 242 feet
at that time.
What is the height of the
Sears Tower?
Indirect Measurement
Since the sun’s rays form similar triangles, the following
proportion can be written.
Now substitute the known values and let x be the height
of the Sears Tower.
Substitution
Cross products
Indirect Measurement
Simplify.
Divide each side by 2.
Answer: The Sears Tower is 1452 feet tall.
Interactive Lab:
Cartography and Similarity
INDIRECT MEASUREMENT On her trip along
the East coast, Jennie stops to look at the
tallest lighthouse in the U.S. located at Cape
Hatteras, North Carolina. At that particular
time of day, Jennie measures her shadow to
be 1 feet 6 inches in length and the length of
the shadow of the lighthouse to be 53 feet 6
inches. Jennie knows that her height is 5 feet
6 inches. What is the height of the Cape
Hatteras lighthouse to the
nearest foot?
A.
196 ft
B. 39 ft
C.
441 ft
D. 89 ft
Lesson 3 CYP3