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

8

1

12

6 x

48 12( 1)

48 12 12

60 12

5

x

x

x

x

.3.92

.5

.36

A

B

C

Bellwork Solution

2291 yx

1 9

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

Example

. ,

. ,

. ,

.

AYes SSS

B Yes SAS

C Yes AA

D No

Are these two triangles similar?

15

9 13

19

Example

. ,

. ,

. ,

.

AYes SSS

B Yes SAS

C Yes AA

D No

Are these two triangles similar?

18

9

7

7

Example

. ,

. ,

. ,

.

AYes SSS

B Yes SAS

C Yes AA

D No

Are these two triangles similar?

A

B

C

D E

bisects

bisects

AC BD

BC ABE

43

21.5

Example

. ~

. ~

. ~

.

A ABC JKL

B ABC OMN

C OMN JKL

D None

Which two, if any, of these triangles are similar

A

12

B

C

J

K

L

M

N

O

9

10

24

18

21

6

7

8

Example

. ~

. ~

. ~

.

A ABC JKL

B ABC OMN

C OMN JKL

D None

Which two, if any, of these triangles are similar

A

16

B

C

J

K

LM

NO

8 18 26

613

Ways to use the theorem

.8

.9

.10.5

.15

A

B

C

D

What value of x makes the two triangles similar

20

12

x+630

3(x-2)

21

Example

.

.

.

AYes

B No

C Sometimes

You enlarge triangle XYW to triangle JHK as shown from vanishing point P. Are the two triangles similar?

P

XJ

W

K

H

XJ=13JW=3WK=21YW=18HK=27

Y

75o75o

Homework

6.5 3-12, 18-23

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