Agenda• 1) Bell Work• 2) Outcomes• 3) Finish 8.4 and 8.5 Notes• 4) 8.6 -Triangle proportionality theorems• 5) Exit Quiz• 6) Start IP

Bell Work• 1) Are the triangles similar? If so, explain

why.

• 2) Solve for the missing variable• a) b)

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

Outcomes• I will be able to:

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

• 2) Understand intersections of parallel lines and triangles

• 3) Use proportionality theorems to find missing lengths

Socrative Student Review• 1) Are the following triangles similar? • A) Yes, by AA• B) Yes, by SAS• C) Yes, by SSS• D) not similar

Socrative Student Review

• 2) Solve for the missing variable if the figures are similar.

• A) 10• B) 12

C) 14• D) 20

Socrative Student Review

• 3) Are the following triangles similar?• A) Yes, by AA• B) Yes, by SAS• C) Yes, by SSS• D) not similar

Socrative Student Review

• 4) Solve for the missing variable if the figures are similar.

• A) 15• B) 30 • C) 36• D) 40

Socrative Student Questions

• 5) Write a statement of proportionality if the figures are similar.

• A)• B)ABCD ~ MNOP• C)• D)

PO

AB

CB

ON

MN

DC

MP

AD

MP

AD

OP

DC

NO

CB

MN

AB

PDOC

NBMA

;

;;

Proportions and Similar Triangles (8.6)

• Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

• Meaning:

TR

QT

UR

SU

Proportions and Similar Triangles

• Converse of the Triangle Proportionality Theorem: If a line divides two sides of a triangle proportionally, then the line is parallel.

• Meaning:

• So, QS is parallel to TUTR

QT

UR

SU

Example

4

2

2.5

x

x

4

2

5.2 X = 3.2

On Your OWN

20

8

25

10

Cross multiply: 200 = 200

So, yes, EB is parallel to DC

20 8

25

10

Use the converse of thetriangle proportionalitytheorem.

Theorem 8.6• Theorem 8.6: If three parallel lines intersect

two transversals, then they divide the transversals proportionally.

• Meaning:XZ

VX

WY

UW

Theorem 8.7• Theorem 8.7: If a ray bisects an angle of a

triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other to sides

• Meaning:

BC

AC

DB

AD

Examples

2.4 1.42.2

2.2x

y

z

On Your OWN

• Find the value of x

9

102

6

4 x

36 = 12x – 60X = 8

How do we set this up?

Example

x

14 - x

9

15

14

xx

How do we set this up?

On Your OWN

• Solve for p

Exit Quiz• 1) Solve for the variable

• 2) Solve for the variable

IP

• Worksheet over triangle proportionality• Due Friday• Also, Friday, we will go over the quizzes

from last week

Chapter 8 Review (Ratios)• ***Remember, in order to have a ratio, we

must make sure that each part of the ratio is in the same units

• Example:

• Remember to always move to the smallest unit

5

6

in

ft

5 5

6 12 72

in in

ft in in

Chapter 8 Review (proportions)

• Remember, proportion rules• If then…

• 1) Reciprocal property

• 2)

• 3)

a c

b d

b d

a c

a b c d

b d

a b

c d

Chapter 8 Review (geometric mean)

• Geometric mean:

• x =

• Example: Find the geometric mean of 6 and 8

• x = = ***Remember to reduce (see board)

a x

x b

a b

6 8 48

Chapter 8 Review (Similar Figures)• Similar Figures must have congruent

angles and side ratios that are proportional

• Similarity Statement:

• Statement of Proportionality

AB BC CD AD

EF FG GH EH

EFGHABCD ~

Chapter 8 Review (Similar Figures)

• Scale Factor: The ratio of corresponding sides in similar figures

• ***Used to find all the missing pieces in similar figures

• ***May be used to find the perimeter of similar figures

• See examples throughout notes

Chapter 8 Review (Similar Triangles)• There are three ways to prove triangles

are similar:• AA – When you know two angles from the

two triangles are congruent to each other

• Example:• Each triangle has

a right angle• And vertical angles

are congruent Triangles similar by AA

Chapter 8 Review (Similar Triangles)• SSS – When all three side ratios are equal

to each other

• Do these ratios equal each other?• Yes, similar by SSS

6 7 8

12 14 16

Chapter 8 Review (Similar Triangles)• SAS – When one angle is congruent and

the side ratios of that included angle are equal to each other

• We know one angle is congruent.

• Check side lengths:

• Yes, similar by SAS7 8

14 16