Transcript Showing Triangles are Similar : AA, SSS and SAS

Showing Triangles are
Similar: AA, SSS and
SAS
Sections 7.3 &7.4
Objectives
• Identify similar triangles using the
AA Similarity Postulate and the SSS
and SAS Similarity Theorems
(similarity shortcuts).
• Use similar triangles to solve
problems.
Key Vocabulary
• None
Postulates
• 15 Angle – Angle (AA) Similarity
Postulate
Theorems
• 7.2 Side – Side – Side (SSS)
Similarity Theorem
• 7.3 Side – Angle – Side (SAS)
Similarity Theorem
Similar Triangles
Are the triangles below similar?
8
4
6
3
37
53
5
10
Do you really have to check all the sides and angles?
NO, there are Shortcuts for determining Similarity.
TRIANGLE SIMILARITY
SHORTCUTS
Postulate 15 Angle-Angle (AA)
Similarity
Angle – Angle (AA) Similarity
Postulate
If two angles of one triangle
are congruent to two angles
of another triangle, then the
two triangles are similar.
Example: If ∠K≅∠Y and
∠J≅∠X, then ∆JKL∼∆XYZ.
Example 1a
A. Determine whether the triangles are similar. If so,
write a similarity statement. Explain your reasoning.
Example 1a
Since mB = mD, B ≅ D
By the Triangle Sum Theorem, 42 + 58 + mA = 180,
so mA = 80.
Since mE = 80, A ≅ E.
Answer: So, ΔABC ~ ΔEDF by the AA Similarity.
Example 1b
B. Determine whether the triangles are similar. If so,
write a similarity statement. Explain your reasoning.
Example 1b
QXP ≅ NXM by the Vertical Angles Theorem.
Since QP || MN, Q ≅ N.
Answer: So, ΔQXP ~ ΔNXM by the AA Similarity.
Your Turn:
A. Determine whether the
triangles are similar. If so, write a
similarity statement.
A. Yes; ΔABC ~ ΔFGH
B. Yes; ΔABC ~ ΔGFH
C. Yes; ΔABC ~ ΔHFG
D. No; the triangles are not
similar.
Your Turn:
B. Determine whether the
triangles are similar. If so, write a
similarity statement.
A. Yes; ΔWVZ ~ ΔYVX
B. Yes; ΔWVZ ~ ΔXVY
C. Yes; ΔWVZ ~ ΔXYV
D. No; the triangles are not
similar.
Thales
The Greek mathematician Thales
was the first to measure the
height of a pyramid by using
geometry. He showed that the
ratio of a pyramid to a staff was
equal to the ratio of one shadow
to another.
Example 2
If the shadow of the pyramid is 576 feet, the
shadow of the staff is 6 feet, and the
height of the staff is 5 feet.
Explain why Thales’ method worked to find
the height of the pyramid?
Find the height of the pyramid?
Example 2
• Explain why Thales’ method worked to find
the height of the pyramid?
• The Triangles formed by the pyramid and
its shadow, and the staff and its shadow
are similar triangles by the AA Postulate.
• Find the height of the pyramid?
AB BC

EF FD
x 576

5
6
6 x  5(576)
6 x  2880
x  480
Similarity proportion
Substitution
Cross product
Simplify
Divide by 6
• Solution: The pyramid is 480 ft tall.
Example 1
Use the AA Similarity Postulate
Determine whether the triangles are
similar. If they are similar, write a
similarity statement. Explain your
reasoning.
SOLUTION
If two pairs of angles are congruent, then the triangles
are similar.
1. G  L because they are both marked as right angles.
Example 1
Use the AA Similarity Postulate
2. Find mF to determine whether F is congruent to J.
mF + 90° + 61° = 180°
mF + 151° = 180°
mF = 29°
Triangle Sum Theorem
Add.
Subtract 151° from each side.
Both F and J measure 29°, so F  J.
ANSWER By the AA Similarity Postulate, FGH ~ JLK
Example 2
Use the AA Similarity Postulate
Are you given enough information
to show that RST is similar to
RUV? Explain your reasoning.
SOLUTION
Redraw the diagram as two triangles: RUV and RST.
From the diagram, you know that both RST and RUV
measure 48°, so RST  RUV. Also, R  R by the
Reflexive Property of Congruence. By the AA Similarity
Postulate, RST ~ RUV.
Your Turn
Use the AA Similarity Postulate
Determine whether the triangles are similar. If they are
similar, write a similarity statement.
1.
ANSWER
yes; RST ~ MNL
2.
ANSWER
yes; GLH ~ GKJ
Your Turn
Use Similar Triangles
Write a similarity statement for the triangles. Then find the value of the
variable.
3.
ANSWER
ABC ~ DEF; 9
4.
ANSWER
ABD ~ EBC; 3
Assignment 7.3
• Pg 375 – 378: #1 – 45 odd
Theorem 7.2 Side-Side-Side
(SSS) Similarity
Side – Side – Side (SSS) Similarity
Theorem:
If the corresponding side lengths of two
triangles are proportional, then the two
triangles are similar.
Example:
Theorem 7.3 Side-Angle-Side
(SAS) Similarity
Side – Angle Side (SAS)
Similarity Theorem:
If two sides of one triangle are
proportional to two sides of
another triangle and the
included angles are
congruent, then the two
triangles are similar.
Example:
Review - Similar Triangles
• Previously, we learned how to determine if two
triangles were congruent (SSS, SAS, ASA, AAS).
There are also several tests to prove triangles are
similar.
• Postulate 15 – AA Similarity
2 s of a Δ are  to 2 s of another Δ
• Theorem 7.2 – SSS Similarity
3 corresponding sides of 2 Δs are proportional
• Theorem 7.3 – SAS Similarity
2 corresponding sides of 2 Δs are proportional and
the included s are 
Example 3a
A. Determine whether the
triangles are similar. If so,
write a similarity statement.
Explain your reasoning.
Answer: So, ΔABC ~ ΔDEC by the SSS Similarity
Theorem.
Example 3b
B. Determine whether the
triangles are similar. If so, write a
similarity statement. Explain
your reasoning.
By the Reflexive Property, M  M.
Answer: Since the lengths of the sides that include
M are proportional, ΔMNP ~ ΔMRS by the SAS
Similarity Theorem.
Your Turn:
A. Determine whether the triangles
are similar. If so, choose the correct
similarity statement to match the
given data.
A. ΔPQR ~ ΔSTR by SSS
Similarity Theorem
B. ΔPQR ~ ΔSTR by SAS
Similarity Theorem
C. ΔPQR ~ ΔSTR by AAA
Similarity Theorem
D. The triangles are not similar.
Example 3b
B. Determine whether the triangles
are similar. If so, choose the correct
similarity statement to match the
given data.
A. The triangles are not similar.
B. ΔAFE ~ ΔACB by SSS
Similarity Theorem
C. ΔAFE ~ ΔAFC by SSS
Similarity Theorem
D. ΔAFE ~ ΔBCA by SSS
Similarity Theorem
Example 4:
In the figure,
and
Determine which triangles in the figure
are similar.
Example 4:
by the Alternate Interior
Angles Theorem.
Vertical angles are congruent,
Answer: Therefore, by the AA Similarity Theorem,
Your Turn:
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 5:
ALGEBRA Given
QT 2x 10, UT 10, find RQ and QT.
Example 5:
Since
because they are alternate interior angles. By AA Similarity,
Using the definition of similar polygons,
Substitution
Cross products
Example 5:
Distributive Property
Subtract 8x and 30 from each
side.
Divide each side by 2.
Now find RQ and QT.
Answer:
Your Turn:
ALGEBRA Given
and CE x + 2, find AC and CE.
Answer:
Example 6
If ΔRST and ΔXYZ are two triangles such that
RS
___
2 which of the following would be sufficient
= __
XY 3
to prove that the triangles are similar?
A
B
C R  S
D
Example 6
___ = RT
___ = RT
___ , then you know that all the sides are
If RS
XY XZ
XZ
2 . This is
proportional by the same scale factor, __
3
sufficient information by the SSS Similarity Theorem to
determine that the triangles are similar.
Answer: B
Your Turn:
Given ΔABC and ΔDEC, which of the
following would be sufficient
information to prove the triangles are
similar?
___ __
A. AC
= 4
DC 3
B. mA = 2mD
___
___
C. AC
= BC
DC EC
___ __
D. BC
= 4
DC 5
Example 7:
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?
Example 7:
Assuming that 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
Example 7:
Simplify.
Divide each side by 2.
Answer: The Sears Tower is 1452 feet tall.
Your Turn:
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
Example 1
Use the SSS Similarity Theorem
Determine whether the triangles are
similar. If they are similar, write a
similarity statement and find the
scale factor of Triangle B to Triangle A.
SOLUTION
Find the ratios of the corresponding sides.
SU
6
6÷6
1
=
=
=
PR
12
12 ÷ 6
2
UT
5
5÷5
1
=
=
=
RQ
10 ÷ 5
2
10
TS
4
4÷4
1
=
=
=
QP
8
÷
4
2
8
All three ratios are equal.
So, the corresponding
sides of the triangles are
proportional.
Example 1
Use the SSS Similarity Theorem
ANSWER
By the SSS Similarity Theorem, PQR ~ STU.
1
The scale factor of Triangle B to Triangle A is .
2
Example 2
Use the SSS Similarity Theorem
Is either DEF or GHJ similar to ABC?
SOLUTION
1. Look at the ratios of corresponding sides in ABC
and DEF.
Shortest sides
DE 4 2
=
AB 6 = 3
ANSWER
Longest sides
FD 8 2
=
CA 12 = 3
Remaining sides
EF 6 2
=
BC 9 = 3
Because all of the ratios are equal,
ABC ~ DEF.
Example 2
Use the SSS Similarity Theorem
2. Look at the ratios of corresponding sides in ABC
and GHJ.
Shortest sides
GH 6 1
=
AB 6 = 1
ANSWER
Longest sides
JG 14 7
=
CA 12 = 6
Remaining sides
HJ 10
BC = 9
Because the ratios are not equal, ABC and
GHJ are not similar.
Your Turn
Use the SSS Similarity Theorem
Determine whether the triangles are similar. If they are
similar, write a similarity statement.
1.
ANSWER
yes; ABC ~ DFE
2.
ANSWER
no
Example 3
Use the SAS Similarity Theorem
Determine whether the triangles
are similar. If they are similar,
write a similarity statement.
SOLUTION
C and F both measure 61°, so C  F.
Compare the ratios of the side lengths that include C
and F.
DF
5
FE 10
5
Shorter sides
Longer sides
=
=
=
AC
CB
6
3
3
The lengths of the sides that include C and F are
proportional.
ANSWER
By the SAS Similarity Theorem,
ABC ~ DEF.
Example 4
Similarity in Overlapping Triangles
Show that VYZ ~ VWX.
SOLUTION
Separate the triangles, VYZ and VWX, and label the
side lengths.
V  V by the Reflexive Property of Congruence.
Shorter sides
Longer sides
VW
4
1
4
=
=
=
VY
4 + 8 12 3
XV
5
1
5
=
=
=
ZV
5 + 10 15 3
Example 4
Similarity in Overlapping Triangles
The lengths of the sides that include V are proportional.
ANSWER
By the SAS Similarity Theorem,
VYZ ~ VWX.
Your Turn
Use the SAS Similarity Theorem
Determine whether the triangles are similar. If they are
similar, write a similarity statement. Explain your
reasoning.
3.
ANSWER
No; H  M but 8 ≠ 12.
6
8
Your Turn
Use the SAS Similarity Theorem
Determine whether the triangles are similar. If they are
similar, write a similarity statement. Explain your
reasoning.
4.
ANSWER
PQ 3
PR 5
1
1
,
Yes; P  P,
and
the
=
=
=
= ;
2
PS
6
2
PT 10
lengths of the sides that include P are proportional,
so PQR ~ PST by the SAS Similarity Theorem.
Summary of Similarity Shortcuts
Assignment 7.4
• Pg. 382 – 385: # 1 – 29 odd, 33 – 37 odd
Joke Time
• Why was the elephant standing on the
marshmallow?
• He didn’t want to fall in the hot chocolate!
• What would you say if someone took your
playing cards?
• dec-a-gon
• What kind of insect is good at math?
• An account-ant