Chapter 7 Similarity and Proportion Express a ratio in simplest form. State and apply the properties of similar polygons. Use the theorems about similar triangles.

7.1 Ratio and Proportion Objectives Express a ratio in simplest form Solve for an unknown in a proportion

Ratio A comparison between numbers 5 : 7 1 2 6 9 3 5 a b This just defines a ratio. Can any of these ratios be simplified? What could the extended ratio be used to do? 5 : 7 : s : 5 : t

Ratio Always reduce ratios to the simplest form This just defines a ratio. Can any of these ratios be simplified? What could the extended ratio be used to do?

Proportion An equation containing ratios Again, just a definition. This does not show how to solve the proportion. Which of these proportions can be solved? Which cannot? Why?

Solving a Proportion First, cross-multiply Next, divide by 5 This shows how to solve a proportion. A proportion can be solved if it is expressed with a single variable.

White Board Practice ABCD is a parallelogram. Find the value of each ratio. A D C B 10 6

White Board Practice AB : BC A D C B 10 6

White Board Practice 5 : 3 A D C B 10 6

White Board Practice BC : AD A D C B 10 6

White Board Practice 1 : 1 A D C B 10 6

White Board Practice m  A : m  C A D C B 10 6

White Board Practice 1 : 1 A D C B 10 6

White Board Practice AB : perimeter of ABCD A D C B 10 6

White Board Practice 5 : 16 A D C B 10 6

White Board Practice x = 2 and y = 3. Write each ratio in simplest form. x to y

White Board Practice x = 2 and y = 3. Write each ratio in simplest form. 2 to 3

White Board Practice x = 2 and y = 3. Write each ratio in simplest form. 6x2 to 12xy

White Board Practice x = 2 and y = 3. Write each ratio in simplest form. 1 to 3

White Board Practice x = 2 and y = 3. Write each ratio in simplest form. y – x x

White Board Practice x = 2 and y = 3. Write each ratio in simplest form. 1 2

7.2 Properties of Proportions Objectives Express a given proportion in an equivalent form.

Means and Extremes a b c d a : b = c : d The extremes of a proportion are the first and last terms The means of a proportion are the middle terms The role of the extremes and means is to make the manipulations learned today easier to describe. = a b c d a : b = c : d

Properties of Proportion is equivalent to 1. 2. 3. Each of these is a different property. 4.

That just means that you can rewrite As any of these 1. 2. 3. Each of these is a different property. 4.

Another Property Select and work several example problems off of the classroom exercises.

White Board Practice If , then 2x = _______

White Board Practice If , then 2x = 28

White Board Practice If 2x = 3y, then

White Board Practice If 2x = 3y, then

White Board Practice If , then

White Board Practice If , then

White Board Practice If , then

White Board Practice If , then

7.3 Similar Polygons Objectives State and apply the properties of similar polygons.

Similar Polygons Same shape Not the same size  Why?

Because then they would be congruent ! Not the same size  Why? Because then they would be congruent !

Similar Polygons (~) All corresponding angles congruent A  A’ B  B’ C  C’ A A’ Write the extended proportion and the congruencies on the diagram. C B B’ C’

Similar Polygons (~) All corresponding sides in proportion AB = BC = CA A’B’ B’C’ C’A’ A A’ Write the extended proportion and the congruencies on the diagram. B C B’ C’

The Scale Factor The reduced ratio between any pair of corresponding sides or the perimeters. 12:3 12 Work several examples of how to find the scale factor, and how to use it to find the unknown parts. 3

Finding Missing Pieces You have to know the scale factor first to find missing pieces. 12 Work several examples of how to find the scale factor, and how to use it to find the unknown parts. 3 10 y

White Board Practice Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find their scale factor A D C B A’ D’ C’ B’ 50 y 30 20 12 x z

White Board Practice 5:3 A D C B A’ D’ C’ B’ 50 y 30 20 12 x z

White Board Practice Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find the values of x, y, and z A D C B A’ D’ C’ B’ 50 y 30 20 12 x z

White Board Practice x = 18 y = 20 z = 13.2 A D C B A’ D’ C’ B’ 50 y 30 20 12 x z

White Board Practice Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find the ratio of the perimeters A D C B A’ D’ C’ B’ 50 y 30 20 12 x z

White Board Practice 5:3 A D C B A’ D’ C’ B’ 50 y 30 20 12 x z

7.4 A Postulate for Similar Triangles Objectives Learn to prove triangles are similar.

AA Simliarity Postulate (AA~ Post) If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. A D This can be shown to prove triangles similar, although there is no proof of it. Why are the sides of a triangle in proportional if the angles are congruent? Do a proof using this postulate. F E B C

Remote Time T – Similar Triangles F – Not Similar

T – Similar Triangles F – Not Similar

T – Similar Triangles F – Not Similar

T – Similar Triangles F – Not Similar

T – Similar Triangles F – Not Similar

7-5: Theorems for Similar Triangles Objectives More ways to prove triangles are similar.

SAS Similarity Theorem (SAS~) If an angle of a triangle is congruent to an angle of another triangle and the sides including those angles are proportional, then the triangles are similar. A D This is a difficult theorem to prove, so it is not wise to prove it in class, unless it is an honors group. Do a proof that uses it, however. Also talk about how this makes the triangles “almost” congruent. Compare the SAS, the SAS and the SAS. F E B C

SSS Similarity Theorem (SSS~) If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar. A D Ditto. F E B C

Homework Set 7.5 7-5 #1-19 odd WS PM 40

7-6: Proportional Lengths Objectives Apply the Triangle Proportionality Theorem and its corollary State and apply the Triangle Angle-bisector Theorem

Divided Proportionally If points are placed on segments AB and CD so that , then we say that these segments are divided proportionally. B This just gets them comfortable with the idea of divided proportionally. Show them several correct proportions that can be written. Also show them an incorrect proportion and why it is incorrect. Most of these are intuitive. D X Y A C See It!

Theorem 7-3 If a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally. Y See It! Ditto. B A X Z

Corollary If three parallel lines intersect two transversals, they they divide the transversal proportionally. R W S X Ditto. T Y See It!

Theorem 7-4 If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides. Y See It! Ditto. W X Z

Construction 12 Given a segment, divide the segment into a given number of congruent segments. Given: Construct: Steps: Do the construction for them on the board. The steps are in the textbook, and can be written down later.

Construction 13 Given three segments, construct a fourth segment so that the four segments are proportional. Given: Construct: Steps: Do the construction for them on the board. The steps are in the textbook, and can be written down later.

Homework Set 7.6 WS PM 41 WS Constructions 12 and 13 7-6 #1-23 odd Quiz next class day