bellwork determine whether the two triangles are similar determine whether the two triangles are...
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Bellwork Determine whether the two triangles are similar
Set 1 ΔABC: mA=90o, mB=44o
ΔDEF: mD=90o, mF=46o
Set 2 ΔABC: mA=132o, mB=24o
ΔDEF: mD=90o, mF=24o
Solve for x
Sun-Yung Alice Chang is a Chinese-American woman who earned a Ph.D. in mathematics from the University of California, Berkley in 1974. In 1995 she won a prize for outstanding research in mathematics. She was born in the year whose sum of digits is 22 and where the units digit is twice the tens digit. What year was she born?
8
1
12
6 x
Clickers
Bellwork Solution
Determine whether the two triangles are similar Set 1
ΔABC: mA=90o, mB=44o
ΔDEF: mD=90o, mF=46o
A B
C
E
F
90 44
46
D 90
46
.
.
.
AYes
B No
C Sometimes
Bellwork Solution
Determine whether the two triangles are similar Set 2
ΔABC: mA=132o, mB=24o
ΔDEF: mD=90o, mF=24o
A B
C
E
F
132 24
24
D 90
24
.
.
.
AYes
B No
C Sometimes
Bellwork Solution
Solve
48 12( 1)
48 12 12
60 12
5
x
x
x
x
8
1
12
6 x
.3.92
.5
.36
A
B
C
Bellwork Solution
1 9
2291 yx yx 2
x y
1 9 2 22
10 3 22
3 12
4
2 8
x x
x
x
x
y x
4 8
She was born in the year whose sum of digits is 22 and where the units digit is twice the tens digit. What year was she born?
.1924
.1936
.1944
.1948
A
B
C
D
Prove Triangles Similar by SSS and SAS
Section 6.5
The Concept Yesterday we looked at looked at how we can prove two
triangles similar by way of looking at their angles Today we’re going to see how we can utilize some of our
congruence methodologies to also prove similarity
TheoremsWhen we studied triangle congruence we used this postulate
Postulate 19: Side-Side-Side Congruence PostulateIf three sides of one triangle are congruent to three sides of a second
triangle, then the two triangles are congruent…
This postulate now becomes
Theorem 6.2: Side-Side-Side Similarity TheoremIf the corresponding side lengths of two triangles are proportional,
then the triangles are similar.
Theorem in actionLet’s look at an example to illustrate this theorem
Are these two triangles similar?
12
8
1018 15
12
What about these two?
12
8
10
10
1412
TheoremsWhen we studied triangle congruence we also saw this postulate
Postulate 20: Side-Angle-Side Congruence PostulateIf two sides and the included angle of one triangle are congruent
to two sides and the included angle of a second triangle, then the two triangles are congruent
This postulate now becomes
Theorem 6.3: Side-Angle-Side Similarity TheoremIf an 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
Theorem in actionLet’s look at an example to illustrate this theorem
Are these two triangles similar?
10
15
15
22.5
ExampleAre these two triangles similar?
12
8
924
16
18
. ,
. ,
. ,
.
AYes SSS
B Yes SAS
C Yes AA
D No
ExampleAre these two triangles similar?
15
9 13
19
. ,
. ,
. ,
.
AYes SSS
B Yes SAS
C Yes AA
D No
ExampleAre these two triangles similar?
18
9
7
7
. ,
. ,
. ,
.
AYes SSS
B Yes SAS
C Yes AA
D No
ExampleAre these two triangles similar?
A
. ,
. ,
. ,
.
AYes SSS
B Yes SAS
C Yes AA
D No
B
C
D E
bisects
bisects
AC BD
BC ABE
43
21.5
ExampleWhich two, if any, of these triangles are similar
A
. ~
. ~
. ~
.
A ABC JKL
B ABC OMN
C OMN JKL
D None
12
B
C
J
K
L
M
N
O
9
10
24
18
21
6
7
8
ExampleWhich two, if any, of these triangles are similar
A
. ~
. ~
. ~
.
A ABC JKL
B ABC OMN
C OMN JKL
D None
16
B
C
J
K
LM
NO
8 18 26
613
Ways to use the theoremWhat value of x makes the two triangles similar
20
12
x+630
3(x-2)
21
.8
.9
.10.5
.15
A
B
C
D
ExampleYou enlarge triangle XYW to triangle JHK as shown from vanishing point P. Are the two triangles similar?
.
.
.
AYes
B No
C Sometimes
P
XJ
W
K
H
XW=16JK=24YW=18HK=27
Y
75o75o
Homework
6.5 1, 4-32 even
HW
25
52
# 4
.
.
.2
A
B
C
# 6
.
.
.
A No
B JKL
C RST
#10
.
.
.
a
A SAS
B SSS
C AA
#10
.
.
.
.
b
A ZXY JDG
B XZY JDG
C ZYX JDG
D YXZ JDG
Most Important Points SSS Similarity Theorem SAS Similarity Theorem