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Chapter 7Similarity 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

5 7 s 5 t

Ratio

- Always reduce ratios to the simplest form

Proportion

- An equation containing ratios

Solving a Proportion

First, cross-multiply

Next, divide by 5

White Board Practice

- ABCD is a parallelogram. Find the value of each

ratio.

White Board Practice

- AB BC

White Board Practice

- 5 3

White Board Practice

- BC AD

White Board Practice

- 1 1

White Board Practice

- m ? A m ? C

White Board Practice

- 1 1

White Board Practice

- AB perimeter of ABCD

White Board Practice

- 5 16

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

- The extremes of a proportion are the first and

last terms - The means of a proportion are the middle terms

a b c d

Properties of Proportion

is equivalent to

1.

3.

2.

4.

That just means that you can rewrite

As any of these

1.

3.

2.

4.

Another Property

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?

- 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

C

B

C

B

Similar Polygons ()

- All corresponding sides in proportion
- AB BC CA
- AB BC CA

A

A

B

C

C

B

The Scale Factor

- The reduced ratio between any pair of

corresponding sides or the perimeters. - 123

12

3

Finding Missing Pieces

- You have to know the scale factor first to find

missing pieces.

12

3

10

y

White Board Practice

- Quadrilateral ABCD Quadrilateral ABCD.

Find their scale factor

White Board Practice

- 53

White Board Practice

- Quadrilateral ABCD Quadrilateral ABCD.

Find the values of x, y, and z

White Board Practice

- x 18
- y 20
- z 13.2

White Board Practice

- Quadrilateral ABCD Quadrilateral ABCD.

Find the ratio of the perimeters

White Board Practice

- 53

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

F

E

B

C

Remote Time

- T Similar Triangles
- F Not Similar

T Similar TrianglesF Not Similar

T Similar TrianglesF Not Similar

T Similar TrianglesF Not Similar

T Similar TrianglesF 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

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

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

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!

B

A

X

Z

Corollary

- If three parallel lines intersect two

transversals, they they divide the transversal

proportionally.

R

W

S

X

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!

W

X

Z

Construction 12

Given a segment, divide the segment into a given

number of congruent segments. Given Construct S

teps

Construction 13

Given three segments, construct a fourth segment

so that the four segments are proportional. Given

Construct Steps

Homework Set 7.6

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