Warm-UpUse the following equation to write three

more true equations:

10

2

25

5

Ratios and ProportionsWhat is a ratio?

A comparison of two numbers by divisionWhat is a proportion?

Two ratios set equal to each other

18

12

3x

y

7

3

10

Words to Algebra:http://www.purplemath.com/modules/translat.htm - under Links on my web page

Addition: increased by more than combined, together total of Sum added to

Words to Algebra:Subtraction

decreased by minus, less difference between less than, fewer than

Words to Algebra:Multiplication

of times, multiplied by product of increased/decreased by a factor of (this

type can involve both addition or subtraction and multiplication!)

Words to Algebra:Division

per, a (miles per gallon, or miles a gallon) out of ratio of, quotient of percent (divide by 100)

Words to Algebra:Equals

is, are, was, were, will be gives, yields sold for same as

Perimeter is the distance around

Example It takes 4 cups of flour to make one

recipe of my favorite chocolate chip cookies. The recipe makes 3 dozen cookies. How much flour is needed to make 42 cookies?

Example: Area of a Rectangle = b * hYou have a rectangle in which the

length of the longest side is twice the length of the shortest side plus two inches, and the ratio of the longest side to the shortest side is 16 to 3. Find the area and the perimeter of the rectangle.

Define:

1. Polygon Closed planar figure with all straight sides

2. Triangle Three sided polygon

3. Congruence Same size and shape

4. Congruence of triangles Triangles that are the same size and

shape

Define:

5. Similarity Same shape, probably not the same size All corresponding angles must be congruent. Corresponding linear dimensions (sides are

just a few of the linear dimensions) must be proportional (they make the same ratio).

The common ratio of the corresponding linear dimensions is called the Scale Factor.

The tilde symbol (~) is used to write a similarity statement

Define:

6. Similarity Statement A statement that identifies which angle are

congruent and which sides have the same ratio (scale factor) in two similar polygons

ORDER IS IMPORTANT!!!!!! Example: CBA ~ ZYX Once you know the triangles are similar, you

can write the proportionality statement (next slide)

Define:7. Proportional Statement

The equations that identify the specific ratios that have to be the same.

That ratio is the scale factor Example for CBA ~ ZYX:

factorscaleZX

CA

YX

BA

ZY

CB_

Define:7. Proportional Statement

Example for CBA ~ ZYX:

Note: The numerator is all parts of CBA, and the denominator is all parts of ZYX.

Please write your proportionality statement comparing corresponding parts of the triangles.

factorscaleZX

CA

YX

BA

ZY

CB_

Example:Given ABC DEF, and AB = 10 BC

= 6, EF = 9, and DF = 12. Write the proportional statement.Find the scale factor of the two trianglesFind the values of DE and ACAre these triangles right triangles?How does the ratio of the area of ABC

compare area of DEF?

Do # 3 – 7 of page one of the packet

Warm UpSolve for x and y

13

5

3

15.2

2

7

y

x

Prove Triangles are Similar:

1. Side-Side-Side (SSS) You need the measures of the length of

ALL SIX SIDES The scale factor of corresponding sides

has to be the same Ratio longest to longest, shortest to

shortest and middle to middle lengths

2. Angle-Angle (AA) You need at least the measure of two

angles in each triangle Two sets of angles must be congruent

To Prove Triangle Similarity3. SAS, Need:

Sides have to make the angle, i.e., the angle has to be between the sides

The angles have to be congruent The sides have to have same scale factor

How to know if Angles are congruent:

1. Could be given by stating the value, use of symbols, or similarity statement.

2. Angles are congruent if they are the same angle in two triangles – think of “stacking” triangles.

3. Vertical AnglesFormed by intersecting lines

How to know if Angles are congruent:

4. Corresponding angles of parallel lines and transversal All acute angle are congruent All obtuse angles are congruent All 90 angles are congruent This could be stacked traingles or triangles

off of a transversal with parallel lines.

Miscellaneous: 1. Angles in a triangle add to be 180

2. The slope between any two points on a straight line is a constant

3. Use Pythagorean Theorem to prove triangles are right triangles or not.

Process to Solve Similar Triangle Problems

1. Prove the triangles are similar by SSS, AA, or SAS (if possible).

2. Write the Similarity Statement

3. Write the Proportional Statement

4. Substitute numbers

5. Solve for variables

6. Check answer(s)

Which sides are corresponding?The following triangles are similar Identify which angle is congruent to

angle A in each situation

AB || CD

EA

B

C

D

AB is not || CD

EA

B

C

D

Determine if these triangles are similar, explain why, write similarity statement, write proportionality statement, and find the missing values

1.6

13

13.5

AB || CD and AD = 19.5

E

A B

C D

Warm UpFive people have been working to build

a patio, but they could not all work the same amount of time. Assume the ratios of the time they spent on the job was 5:5:6:6:8, and assume they worked a total of 360 hours. How many hours did each work?

ExamplesSolve for x and y

Prove if similar and write similarity statement and proportionality statement:

13

5

3

15.2

2

7

y

x

1.82

5.74

4.1

1.3A

D

E

B

C

BC ll DEShow triangle ADE is Similar to triangle ABC

C

D

A

E

B

AD = DE = AEAB BC AC

The ratio of any corresponding pair of sidesis the same

BC ll DEShow triangle ADE is Similar to triangle ABC

C

D

A

E

B

Use:AD = AEAB AC

Write AD and AE as the sum of the partsand simplify

Can you proveBD = CEAB AC

C

D

A

E

B

BD = CEAB AC

If a line is parallel to one side of a triangleand intersects the other two sides, then itseparates the sides into segments ofproportional lengths!

COOL!!!

C

D

A

E

B

ABC ~ DEF, AB = 7, BC = 9, AC = 12 and EF = 12.

Find the perimeter of DEF

BCDE2

1

What can we sayabout the ratio of BCto DE if B and C arethe midpoints of ADand AE? C

D

A

E

B

Summary3: In two similar triangles, the ratio of two sides of one triangle is the same as the ratio of the corresponding sides of the other triangle.

This comes from the similarity statement:

And the proportional statement.DEFABC ~

rScaleFactoEF

BC

DF

AC

DE

AB

American Choice Lesson 3 Introduction - We discussed:Mathematics words to equations:

“Same as” means equals “of” means multiply

Perimeter is the distance around

Area of a triangle = 0.5bh

Area of a rectangle = bh or lw