Download Section 3

5-Minute Check on Lesson 6-2
Transparency 6-3
1. Determine whether the triangles are similar.
Justify your answer.
Yes: corresponding angles 
corresponding sides have same proportion
2. The quadrilaterals are similar. Write a
similarity statement and find the scale
factor of the larger to the smaller
quadrilateral. ABCD ~ HGFE
Scale factor = 2:3
3. The triangles are similar. Find x and y.
x = 8.5, y = 9.5
4.
Standardized Test Practice:
Which one of the following statements is
always true?
A Two rectangles are similar
B Two right triangles are similar
C Two acute triangles are similar
D Two isosceles right triangles are similar
Click the mouse button or press the
Space Bar to display the answers.
Lesson 6-3
Similar Triangles
Objectives
• Identify similar triangles
• Use similar triangles to solve problems
Vocabulary
• None new
Theorems
• Postulate 6.1: Angle-Angle (AA) Similarity
If two angles of one triangle are congruent to
two angles of another triangle,
then the triangles are similar
• Theorem 6.1: Side-Side-Side (SSS) Similarity
If all the measures of the corresponding
sides of two triangles are proportional,
then the triangles are similar
Theorems Cont
• Theorem 6.2: Side-Angle-Side (SAS) Similarity
If the measures of two sides of a triangle are
proportional to the measures of two corresponding
side of another triangle and the included angles are
congruent, then the triangles are similar
• Theorem 6.3: Similarity of triangles is
reflexive, symmetric, and transitive
– Reflexive: ∆ABC ~ ∆ABC
– Symmetric: If ∆ABC ~ ∆DEF, then ∆DEF ~ ∆ABC
– Transitive: If ∆ABC ~ ∆DEF and ∆DEF ~ ∆GHI, then
∆ABC ~ ∆GHI
AA Triangle Similarity
A
P
Third angle must be congruent as well
(∆ angle sum to 180°)
Q
From Similar Triangles
B
Corresponding Side Scale Equal
C
AC
AB
BC
---- = ---- = ---PQ
PR
RQ
R
If Corresponding Angles Of Two Triangles Are Congruent,
Then The Triangles Are Similar
mA = mP
mB = mR
SSS Triangle Similarity
A
P
From Similar Triangles
Corresponding Angles Congruent
Q
A  P
B  R
C  Q
B
C
R
If All Three Corresponding Sides Of Two Triangles Have Equal Ratios,
Then The Triangles Are Similar
AC
AB
BC
---- = ---- = ---PQ
PR
RQ
SAS Triangle Similarity
A
P
Q
B
C
R
If The Two Corresponding Sides Of Two Triangles Have
Equal Ratios And The Included Angles Of The Two
Triangles Are Congruent, Then The Triangles Are Similar
AC
AB
---- = ---PQ
PR
and A  P
Example 1a
In the figure, AB // DC, BE = 27, DE= 45, AE = 21, and
CE = 35. Determine which triangles in the figure are
similar.
Since AB ‖ DC, then
BAC  DCE by the
Alternate Interior Angles
Theorem.
Vertical angles are congruent,
so BAE  DEC.
Answer: Therefore, by the AA Similarity Theorem,
∆ABE  ∆CDE
Example 1b
In the figure, OW = 7, BW = 9, WT = 17.5, and WI = 22.5.
Determine which triangles in the figure are similar.
I
Answer:
Example 2a
ALGEBRA: Given RS // UT, RS=4, RQ=x+3,
QT=2x+10, UT=10, find RQ and QT
Since
because they are alternate interior angles. By AA Similarity,
Using the definition of similar polygons,
Example 2a cont
Substitution
Cross products
Distributive Property
Subtract 8x and 30 from each side.
Divide each side by 2.
Now find RQ and QT.
Answer:
Example 2b
ALGEBRA Given AB // DE, AB=38.5, DE=11, AC=3x+8, and
CE=x+2, find AC and CE.
Answer:
Example 3a
INDIRECT MEASUREMENT Josh wanted to measure
the height of the Sears Tower in Chicago. He used a 12foot 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?
Assuming that the sun’s rays form similar ∆s,
the following proportion can be written.
Example 3a cont
Now substitute the known values and let x be the height
of the Sears Tower.
Substitution
Cross products
Simplify.
Answer: The Sears Tower is 1452 feet tall.
Example 3b
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?
Answer: 196 ft
Summary & Homework
• Summary:
– AA, SSS and SAS Similarity can all be used to
prove triangles similar
– Similarity of triangles is reflexive, symmetric, and
transitive
• Homework:
– Day 1: pg 301-302: 6-8, 11-15
– Day 2: pg 301-305: 9, 18-21, 31
Ratios:
QUIZ Prep
1)
x
3
=
12
4
2)
x - 12 x + 7
=
-4
6
3)
28
7
=
z
3
4)
14
x+2
=
10
5
Similar Polygons
H
K
A
12
C
J
B
16
x+1
D
C
5
W
D
Similar Triangles (determine if similar and list in proper order)
P
W
E
x+3
A
6
V
35°
C
85°
B
T
40°
N
6
L
x-3
4
G
A
10
y+1 8
B
10
S
Z
Q
F
R
S
1
11x - 2
W
M