Geometry Chapter 7 Ratios & Proportions

Properties of Proportions

Similar Polygons

Similarity Proofs

Triangle Angle Bisector Theorem

1

Name: _________________________________________________________________ Geometry Assignments – Chapter 7

Similar Polygons Date Due

Section

Topics

Assignment

Written Exercises

7.1 &

7.2

Ratio Proportion Means Extreme

Pg. 244 # 25-31 odd AND Pg. 247 #4, 6, 8, 20, 22, 24, 26, 28, 33 & 37

7.3 & 7.4

Similarity Similar Polygons Scale Factor AA Similarity

Pg. 250 (bottom)-251 # 2-14 even, 15-22, 24-27 AND Pg. 257 #2-20 even

7.5

SAS Similarity Postulate SSS Similarity Postulate

Pg. 266 #2-10 even, 14 AND Pg. 258 # 21-25, 27

7.6

Proportional Lengths Triangle Proportionality

Theorem Triangle Angle-Bisector

Theorem

Pg. 272-273 #2-11, 20, 22, 23, 25

Similarity Proofs

Similarity Proofs Worksheet

Similarity Proofs (cont)

& REVIEW

Similarity Proofs Worksheet

Chapter Test

Remember, if you have any questions or are having difficulty, please come in for extra help.

2

145105

12

15

A B

DC

5x 5 12x 2 3x

92

3

a c

b d

2 8

3 12

a c

b d

ad bca b

c d

b d

a c

a b c d

b d

4

A

BC

G H

5 35

24 9

x

5.4 6

7.2x

4 3

9 5

6 x

14

56

x

x

3 7

4 5x x

2 5 2 1

8 4

x x

AG AH

GB HC

5

Ratio and Proportion WS Name __________________________

Geometry – 7-1 and 7-2 Date _______________ Block ______

Solve each proportion for x.

1. 6 27

24 9

x 2.

4.8 6

8.4x 3.

2 7

5 10

8 x

4. 6

150

x

x 5.

3 7

4 4x x

6.

5 2 3 1

8 4

x x

7. 22

18 12

x 8.

22 2

18x 9.

51

84

16 x

10. 8

50

x

x 11.

3 5

3 1x x

12.

3 7 2 1

15 21

x x

6

________________________________________________________________

Complete each proportion. [Use properties!]

13. Given: 9

17

w

x 14. Given: 6 : : 7y h

a) 9

w a) y h

b) x

w b)

6

h

c) 9x c) 6 y

y

d) w x

x

d)

6

y

__________________________________________________________________

15. Three numbers aren’t known, but the ratio of the numbers is 1 : 3 : 8.

Is it possible that the numbers are:

a) 1, 3, and 8?

b) 3, 9, and 21?

c) 10, 30, and 80?

d) y, 3y, and 8y?

e) x, 3y, and 8z?

____________________________________________________________________

More practice:

p. 246 Class Exercises (1-12) Challenge: 13, 14

7

A B

C

D

ET

P

Q

RS

GEOMETRY – Notes 7.3 DATE: ____________________________

LOOKING BACK…

Given the statement, “If today is Monday, then I have school.”

Write a. the contrapositive

b. the inverse

c. the converse

Name the quadrilateral:

I have four congruent sides and no right angles. _____________________

I have two congruent segments and the other pair of sides are parallel. ______________

Simplify

18

36

a

9 6

3

x y

2 2

5 5x y

x y

NOTES – 7.3 – SIMILAR POLYGONS

Two polygons are similar if their vertices can be paired so that:

a.

b.

Notation

If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the

________________________________.

For the example on the page before, the ________________________ is =

The ratio of the perimeters of two similar polygons is ________________________________

____________________________________________________________________________.

8

Similar Polygons WS1 [notes] Name ______________________

Geometry – section 7-3 Date _________ Block ______

Determine whether the polygons are similar. Explain your reasoning.

In each question, the given polygons are similar. Find the value of x.

5. 6.

7. 8.

9

In each exercise below, determine whether the polygons are similar. Explain

your reasoning. If the polygons are similar, write a similarity statement.

9. 10.

11. 12.

Complete each of the following.

13. 14.

15. An architect is making plans for a rectangular office building that is 840 feet

long and 252 feet wide. A blueprint of the floor plan for the first floor is 15 inches

long. How wide is the blueprint?

____________________________________________________________________________

10

p.250 CE (1-9) Are the quadrilaterals similar? If they aren't, tell why not.

1. ABCD and EFGH 2. ABCD and JKLM

3. ABCD and NOPQ 4. JKLM and NOPQ

____________________________________________________________________________

5. If the corresponding angles of two polygons are congruent, must the polygons be similar?

6. If the corresponding sides of two polygons are in proportion, must the polygons be similar?

7. Two polygons are similar. Do they have to be congruent?

8. Two polygons are congruent. Do they have to be similar?

9. Are all regular pentagons similar? Why?

___________________________________________________________

10. JUDY ~ J'U'D'Y'. Complete.

a) m<Y' = _____ and m<D = _____

b) The scale factor of JUDY to J'U'D'Y' is _______

c) Find DU, Y'J' , and J'U'.

d) The ratio of perimeters is ______

e) Explain why it is not true that DUJY ~ Y'J'U'D'.

__________________________________________________________

11

A

B

C

H

I

G

7-4 Notes: Similarity Proof

Similar Polygons: Corresponding angles of similar polygons are congruent AND

corresponding sides are proportional.

______________________________________________________________________

Postulate – If two angles of one triangle are congruent to two angles of another triangle, then the

two triangles are similar.

Called –

Example: Given :B I

A G

We can conclude:

12

Algebra first….

13

Similar Triangles WS2 Name __________________________

Geometry – 7-3 & 7-6 Date _____________ Block ________

________________________________________________________________________

Part 1: Use properties of similar triangles to set up equations and solve as

needed. Show your work!

1.

2.

3.

4.

14

5.

6.

7.

________________________________________________________________________

Part 2: Proving the Triangle Proportionality Theorem. Fill in the following proof…

Statements Reasons .

1) DE BC given

2) ADE ABC and AED ACB

3) ABC ADE AA similarity

4) AB CA

DA EA

5) AD + DB = AB; AE + EC = AC

6) AD DB AE EC

DA EA

7) BD CE

DA EA

If a line parallel to one side of a triangle intersects the other two sides, then it

divides those sides proportionally.

15

________________________________________________________________________

Part 3: Use similar triangles or the new theorem from part 2 (triangle

proportionality theorem) to solve for the value of x in each question.

16

2)

Given: <QVR ≅ <S

Prove: (QR)(ST) = (QT)(VR)

17

18

p.271 CE (3-5). Write a proportion for each question. Solve question 4 & 5.

p.272 WE(6-8). Find the value of x.

19

E

A B

D C

F

B C

E

A D

C

A B

DE

7-4 & 7-5 Triangle Similarity Proofs

AA Similarity Theorem:

SSS Similarity Theorem:

SAS Similarity Theorem:

_____________________________________________________________________________

Similarity Proofs WS1 Name _____________________________

Geometry Date _________ Block ___________

1. Given: AB DC

Prove: AE DE CE BE

2. Given: ABCD is a parallelogram

Prove: AF EF DF BF

3. Given: ; midpoint of ; midpoint ofABC D AC E BC

Prove: CD DE

CA AB

[hint : segment DE must be a midsegment…]

20

246

x

B

C

AD

30

x

A

B C

D

x + 6x

x + 1

2

x

6

15

10

A

B P

D

E C

4. Given: ; ;ABC with AB AC PD AB

AE BC

Prove: EC AC

DB PB

________________________________________________________________________

5. Find the value of x.

given: ;BC CA CD BA

and lengths in diagram

[use similar triangles]

6. If the ratio of the lengths of the segments formed on a hypotenuse of a right

triangle from the intersection of the altitude is 1:9; and the length of the altitude

to the hypotenuse is 6, find the lengths of the two segments formed on the

hypotenuse. [use similar triangles]

7. Find x.

given: ; ; 50BD AC AB BC AC

and lengths in diagram

[use similar triangles]

8. Find x.

given: the two horizontal segments

are parallel; not to scale!

9. Find x.

given: the three vertical segments are

parallel

21

Similarity Proof WS2 Name _________________________

Geometry Date _____________ Block ______

______________________________________________________________________

22

__________________________________________________________________

23

______________________________________________________________________

24

X

D

C

B

A

W

Z

X Y

V

R

Q

P

ST

D

CA

B

E

D

B

A C

E

Similarity Proofs WS3 Name ____________________________

Geometry Date ________________ Block ______

1. Given: A C 2. Given: VZ WZ

XZ YZ

Prove: ABX CDX Prove: 1 2

3. Given: PST PRQ 4. Given:

;

;

BA BC

AE BC

BD AC

Prove: PS PT

PR PQ Prove:

AC AE

BA BD

5. Given: EDA DAC 6. Given: 1 A 7. Given: 1 2

Prove: BED BAC Prove: ABE CDE Prove: MLN JLK

E

C

A

B

D

1

N

KJ

L

M 1

2

25

Chapter 7 Review Name __________________________

Geometry Date __________ Block _______

1. Two angles are complementary and are in the ratio of 7:8. Find the value of the

smaller angle.

2. Quad ABCD with m<A: m<B: m<C : m<D = 7:2:2:7. Find the measure of all the

angles in the quadrilateral and state what special type of quadrilateral ABCD is.

3. A hexagons angles are in the ratio of 4 : 3 : 7 : 3 : 6 : 7 , find the measure of

the largest angle.

4. Solve the following proportions for all variables.

a. 9 5

3 6 3x x

b.

12

4 5

x

x

c.

7 13

2 2

x x

x

d. 4

9

x

x e.

3

3 2

y x x and

5

4 5

x y

5. Determine whether the given figures are sometimes, always or never similar.

a. Two Rectangles

b. Two Squares

c. Two isosceles trapezoids with congruent base angles

d. Two regular dodecagons

e. A scalene triangle and an isosceles triangle

f. Two rhombi with at least one < congruent

g. Two triangles with proportional sides

6. ~Quad ABCD Quad HIJK

a. If, AB = 8, JK = 12, and CD = 9, find HI.

b. If m<B = 50 , find m<I.

c. If the perimeter of Quad HIJK is equal to 36, what is the perimeter

of Quad ABCD?

7. If 2500 square feet of grass emits enough oxygen for a family of 4, how much

grass is needed to supply oxygen for a family of five?

26

8. If a car uses 15 gallons of gas to travel 500 miles, how many miles does it get

for one gallon of gas?

9. Prove whether or not the following triangles are similar. Justify your answer.

10. Find x.

11. Find w, x, y, z.

[diagram not to scale!]

12. Use the diagram below to answer each question. Segment AC is an angle

bisector.

a. If BC = 20, AB = 18, AD = 45 find CD.

b. If AB = 12, AD = 15, and BD = 9, find CD.

13. Given: AC AB , BV AB 14. Given: Rectangle ABCD , EFB CGE

Prove: AC JB BV CJ Prove ~ABG DCF

4

3

5

25

15

20

27

15.

16.

17.

18.

Given: Iso. Trap. ABFC with AC BF , //AD BF

Prove: AD DF EF ED

19. Given: 1 2

Prove: A D

28

EXTRA PRACTICE – See me for answers: Pg 252 (Self Test #1) #1-6 Pg 274 (Self Test #2) #1-11 Pg 277-278 (Chpt Rev) #1-24 Pg 279 (Chpt Test) #1-15