Warm-Up  Use the following equation to write three more true equations:

Ratios and Proportions  What is a ratio?  A comparison of two numbers by division  What is a proportion?  Two ratios set equal to each other

Words to Algebra: slat.htmhttp:// slat.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 * h  You 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:

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.

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 triangles  Find the values of DE and AC  Are 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 Up  Solve for x and y

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 Similarity 3. 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 Angles Formed 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 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

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

Warm Up  Five 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?

Examples  Solve for x and y  Prove if similar and write similarity statement and proportionality statement:

  ABC ~  DEF, AB = 7, BC = 9, AC = 12 and EF = 12.  Find the perimeter of DEF

Summary 3: 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.

American ChoiceLesson 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