chapter Review

Connecting BIG ideas and Answering the Essential Questions

1 SimiiarityYou can set up andsolve proportions usingcorresponding sides ofsimilar polygons.

2 Reasoningand Proof

Two triangles are similarif certain relationshipsexist between two

or three pairs ofcorresponding parts.

3 Visualization

Sketch and label trianglesseparately in the sameorientation to see how

the vertices correspond.

Ratios and Proportions (Lesson 7-1)

The Cross Products Property states that if

f = 5, then ad = be

Similar Polygons (Lesson 7-2)

Corresponding angles of similar polygonsare congruent, and corresponding sides ofsimilar polygons are proportional.

Proving Triangles Similar (Lesson 7-3)

Angle-Angle Similarity (AA ~) PostulateSide-Angle-Side Similarity (SAS ~) TheoremSide-Side-Side Similarity (SSS ~) Theorem

Seeing Similar Triangles(Lessons 7-3 and 7-4)

A A

Proportions in Triangles

(Lessons 7-4 and 7-5)

Geometric Means in Right Triangles

Side-Splitter Theorem

§_ _£

b d

B DAABC-AECD

Triangle-Angle-Bisector Theorem

d L = Lb d

Chapter Vocabularyextended proportion (p. 440)extended ratio (p. 433)extremes (p. 434)geometric mean (p. 462)

indirect measurement (p. 454)means (p. 434)proportion (p. 434)ratio (p. 432)

scale drawing (p. 443)scale factor (p. 440)similar figures (p. 440)similar polygons (p. 440)

Choose the correct term to complete each sentence.

1. Two polygons are J_ if their corresponding angles are congruent andcorresponding sides are proportional.

2. A(n) is a statement that two ratios are equal.

3. The ratio of the lengths of corresponding sides of two similar polygons is the

4. The Cross Products Property states that the product of the J_ is equal to theproduct of the ? .

480 Chapter? Chapter Review

7-1 Ratios and Proportionsr

Quick ReviewA ratio is a comparison of two quantities by division. Aproportion is a statement that two ratios are equal. The

Cross Products Property states that if f = where b 0and d 0, then ad = be.

ExampleWhat is the solution of = 4,

a: + 3 6*

6a: = 4{x + 3) Cross Products Property

6a: = 4a: + 12 Distributive Property

2a: =12 Subtract 4x from each side.

X = 6 Divide each side by 2.

Exercises

5. A high school has 16 math teachers for 1856

math students. What is the ratio of math teachers

to math students?

6. The measures of two complementary angles arein the ratio 2 : 3. What is the measure of the

smaller angle?

Algebra Solve each proportion.

II

18

21

8.II

XX4- 410.8 2

35 X4- 9

X

-

7-2 and 7-3 Similar Polygons and Proving Triangles Similar

Quick Review

Similar polygons have congruent corresponding anglesand proportional corresponding sides. You can provetriangles similar with limited information about congruentcorresponding angles and proportional correspondingsides.

What You Need

two pairs of = angles

two pairs of proportional sides

and the included angles =

three pairs of proportional sides

Postulate or Theorem

Angle-Angle (AA ~)

Side-Angle-Side (SAS ~)

Side-Side-Side (SSS ~)

ExampleIs AABC similar to ARQP? How do

you know?

You know that /LA = AR.

^ = ̂ = |, so the trianglesare similar by the SAS ~ Theorem.

B P 2 fi

Exercises

The polygons are similar. Write a similarity statement andgive the scale factor.

11. K 28

24

jd

24

36 NU P

13. City Planning The length of a rectangularplayground in a scale drawing is 12 in. If the scale is1 in. = 10 ft, what is the actual length?

14. Indirect Measurement A 3-ft vertical post casts a24-in. shadow at the same time a pine tree casts a30-ft shadow. How tall is the pine tree?

Are the triangles similar? How do you know?

15. A 8 16. R

' ̂ 3 E

C PowerGeometry.com I Chapter 7 Chapter Reviev7 481

7-4 Similarity in Right Triangles

Quick Review

CD is the altitude to the

hypotenuse of right AABC.

. AABC ~ AACD,

AABC ACBD, and

AACD - ACBD

^_CD ^AB _CB* CD DB'AC DB

ExampleWhat is the value of x?

5-i-x 10

10 5

5(5 + x] = 100

25 + 5X = 100

5X = 75

x= 15

Write a proportion.

Cross Products Property

Distributive Property

Subtract 25 from each side.

Divide each side by 5.

7-5 Proportions in Triangles

Exercises

Find the geometric mean of each pair of numbers.

17. 9 and 16 18. 5 and 12

Algebra Find the value of each variable. Write your

answer in simplest radical form.

19. 12

V J ///

Quick Review

Side-Splitter Theorem and Corollary

If a line parallel to one side of a triangle intersects the othertwo sides, then it divides those sides proportionally. If three

parallel lines intersect two transversals, then the segmentsintercepted on the transversals are proportional.

Triangle-Angle-Bisector Theorem

If a ray bisects an angle of a triangle, then it divides theopposite side into two segments that are proportional to theother two sides of the triangle.

ExampleWhat is the value of xl

12 = 915 X

12x = 135

X= 11.25

Write a proportion.

Cross Products Property

Divide each side by 12.

Exercises

Algebra Find the value ofx.

X- 3

482 Chapter? Chapter Review