Bell Work • 1) A) Write a similarity statement for the similar figures

below:• B) What is the similarity ratio?• C) Solve for x and y

• 2) Simplify the following ratios:• A) B)

• 3) Write a statement of proportionality for the figures below:

ft

in

3

15

ft

yds

24

12

Agenda• 1) Bell Work

• 2) IP Check

• 3) Outcomes

• 6) 8.4 Notes

• 7) IP – Assessment #11

Outcomes

• I will be able to:

• 1) Use ratios and proportions to find missing side lengths of similar figures

• 2) Determine if triangles are similar

• 3) Use similarity shortcuts to determine if triangles are similar

Similar Triangle Activity1. Look at Triangle ABC and Triangle DEF: A (1, 1) and D (1, -1) B (4, 1) E (7, -1) C (4, 5) F (7, -9)

2. Look at Triangle JKL and Triangle MNO: J (1, 2) and M (-1, -1) K (5, 2) N (-3, -1) L (3, -2) O(-2, -3)

3. Look at Triangle PQR and Triangle STV: P (-2, 0) and S (-1, 1) Q (0, 4) T (0, 3) R (2, 0) V (1, 1)

Are these triangles similar? Yes/NoWhy?

Similarity Statement:

____________________ 

Statement of Proportionality?

__________________

What pieces did we need to determine similarity?

Results• 1. Look at Triangle

ABC and Triangle DEF where:

• A (1, 1) and D (1, -1)• B (4, 1) E ((7, -1)• C (4, 5) F (7, -9)• AB = ______ • DE = ______• m∠A = ___, m∠D=

____

• AC = _____ • DF = ______  

• Are these triangles similar? Yes/No

• Why?

• Similarity Statement:

____________________ 

Statement of Proportionality?

__________________

• What pieces did we need to determine similarity?

Results• 2. Look at Triangle

JKL and Triangle MNO where:

• J (1, 2) and M (-1, -1)• K (5, 2) N (-3, -

1)• L (3, -2) O (-2, -

3)_____ and _____ MJ

_____ and _____ NK_____ and _____ OL

• Are these triangles similar? Yes/No

• Why?

• Similarity Statement:

____________________ 

Statement of Proportionality?

__________________

• What pieces did we need to determine similarity?

Results• 3. Look at Triangle

PQR and Triangle STV where:

• P (-2, 0) and S (-1, 1)

• Q (0, 4) and T (0, 3)• R (2, 0) and V (1, 1)

• PQ = ____ ST =_____• QR = ____ TV = ____• RP = _____ VS= ____

• Are these triangles similar? Yes/No

• Why?

• Similarity Statement:

____________________ 

Statement of Proportionality?

__________________

• What pieces did we need to determine similarity?

Why?• What did we find during this activity?

• Did we need to look at every single angle and every single side to determine if the triangles were similar?

• No, so there are triangle similarity shortcuts just as there were triangle congruence shortcuts.

Shortcuts

• Just as when looking at triangle congruence, there are shortcuts to help us determine if triangles are *similar.

Similar Triangles

If Two angles of one triangleare congruent to two anglesin another triangle, then the two triangles are similar.

Similarity Statement:

YXZKJL ~

Why do we only need two angles?

Examples

• 1) Explain why the triangles are similar and write a similarity statement

• Angle A is congruent• to Angle D• Angle C is congruent• in both triangles • because of vertical angles• So, by AA

B

AC

D

EDECABC ~

Examples

• 2) Are the following triangles similar?

• So, yes by the AA • similarity postulate

Sometimes you mayhave to solve for a piece that isn’t given to us in order to determine if they aresimilar.

x

x + 40 + 88 = 180x = 52

On Your Own• 3. Explain why the triangles are similar

and write a similarity statement• ΔRQP ~ ΔVUP by the AA postulate

because:

• Suppose RP = 15, RQ = 10 and UV = 7• Find VP and RV• What is the scale factor?

• Write a proportion to • solve:

PP

UQ

x710

15

10

7

1510

7 x

Similarity Theorems

If all the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.

If 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

1014

18

57

9

18

9

14

7

10

5*Check the side ratios:

612

816 95°

95°

*Check the side ratios and the angle between them

Examples

6

18

4

12

What should we do first?1. Label everything

What do we need to do next?

2. Look at what pieces of each triangle we actually have

We have two sides from eachso we can compare side ratios

12

4

18

6 So, we know two side ratios are the same. What

else do we know?Both triangles have Angle A, so it is congruent

So, these triangles are similar by SAS similarity shortcut

On Your OWN

Examples

60

48

40

52

65

x

1. Label what we know

2. Set up a proportion to find the missing piece.

What do we have to make sure wedo?

Compare pieces of the small trianglewith pieces of the big triangle

x

40

108

48 48x = 4320

x = 90m

Examples

1. Label what we know

2. What can we check?

6.32

6.38

6.27

6.32

22

26

***Be sure to create a proportion in which the corresponding parts line up

On Your Own