Bellwork

BellworkBellwork Determine whether the two triangles are similarDetermine whether the two triangles are similar

Set 1Set 1 ΔΔABC: mABC: mA=90A=90oo, m, mB=44B=44oo

ΔΔDEF: mDEF: mD=90D=90oo, m, mF=46F=46oo

Set 2Set 2 ΔΔABC: mABC: mA=132A=132oo, m, mB=24B=24oo


Solve for xSolve for x

Sun-Yung Alice Chang is a Chinese-American woman who earned a Ph.D. in 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 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 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?is 22 and where the units digit is twice the tens digit. What year was she born?8

1

12

6 x

No Clickers

Bellwork SolutionBellwork Solution

Determine whether the two triangles are similarDetermine whether the two triangles are similar Set 1Set 1

ΔΔABC: mABC: mA=90A=90oo, m, mB=44B=44oo


A B

C

E

F

90 44

46

D 90

46Yes


Determine whether the two triangles are similarDetermine whether the two triangles are similar Set 2Set 2

ΔΔABC: mABC: mA=132A=132oo, m, mB=24B=24oo


A B

C

E

F

132 24

24

D 90

24 No


SolveSolve

48 12( 1)

48 12 12

60 12

5

x

x

x

x

8

1

12

6 x


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 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?twice the tens digit. What year was she born?

Prove Triangles Similar by Prove Triangles Similar by SSS and SASSSS and SAS

Section 6.5Section 6.5

The ConceptThe Concept Yesterday we looked at looked at how we can prove two Yesterday we looked at looked at how we can prove two

triangles similar by way of looking at their anglestriangles similar by way of looking at their angles Today we’re going to see how we can utilize some of our Today we’re going to see how we can utilize some of our

congruence methodologies to also prove similaritycongruence methodologies to also prove similarity

TheoremsTheoremsWhen we studied triangle congruence we used this postulate

Postulate 19: Side-Side-Side Congruence PostulatePostulate 19: Side-Side-Side Congruence PostulateIf three sides of one triangle are congruent to three sides of a second If three sides of one triangle are congruent to three sides of a second

triangle, then the two triangles are congruent…triangle, then the two triangles are congruent…

This postulate now becomes

Theorem 6.2: Side-Side-Side Similarity TheoremTheorem 6.2: Side-Side-Side Similarity TheoremIf the corresponding side lengths of two triangles are proportional, If the corresponding side lengths of two triangles are proportional,

then the triangles are similar.then the triangles are similar.

Theorem in actionTheorem 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

18 15 12

12 10 8

,Yes by SSS

14 12 10

12 10 8

Not similar

TheoremsTheoremsWhen we studied triangle congruence we also saw this postulate

Postulate 20: Side-Angle-Side Congruence PostulatePostulate 20: Side-Angle-Side Congruence PostulateIf two sides and the included angle of one triangle are congruent If 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 to two sides and the included angle of a second triangle, then the two triangles are congruenttriangles are congruent

This postulate now becomes

Theorem 6.3: Side-Angle-Side Similarity TheoremTheorem 6.3: Side-Angle-Side Similarity TheoremIf an angle of one triangle is congruent to an angle of a second If an angle of one triangle is congruent to an angle of a second

triangle and the lengths of the sides including these angles triangle and the lengths of the sides including these angles are proportional, then the triangles are similarare proportional, then the triangles are similar

Theorem in actionTheorem in actionLet’s look at an example to illustrate this theorem

Are these two triangles similar?

10

15

15

22.5

22.5 15

15 10

,Yes by SAS

ExampleExampleAre these two triangles similar?

12

8

924

16

18

,Yes by SSS


15

9 13

19

No


18

9

7

7

No


A

,Yes by SAS

B

C

D E

bisects

bisects

AC BD

BC ABE

43

21.5

ExampleExampleWhich two, if any, of these triangles are similar

A

OMN JKL

12

B

C

J

K

L

M

N

O

9

10

24

18

21

6

7

8

ExampleExampleWhich two, if any, of these triangles are similar

A

None

16

B

C

J

K

LM

NO

8 18 26

613

Ways to use the theoremWays to use the theoremWhat value of x makes the two triangles similar

20

12

x+630

3(x-2)

21

8x 20 6 12

30 21 3( 2)

x

x

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

16 18

24 27,Yes similar

P

XJ

W

K

H

XW=16JK=24YW=18HK=27

Y

75o75o

HomeworkHomework

6.56.5 1, 4-32 even 1, 4-32 even

Most Important PointsMost Important Points SSS Similarity TheoremSSS Similarity Theorem SAS Similarity TheoremSAS Similarity Theorem

Bellwork

Documents

Transcript of Bellwork