Chapter SevenSimilar Polygons

Ruby Weiner & Leigh Zilber

7.1 Ratios and Proportions

• Ratio: the quotient of 2 values A

B

C

D

60

30

90 90

60

30

1) Find the ratio of AE to BE10: 5x 2:x

2) Find the ratio of largest > of triACE to smallest > of triBDE90:30 3:1

E

10

5x

Ratio Practice Problems

• A telephone pole 7 meters tall snaps into 2 parts. The ratio of the 2 parts is 3:2. Find the length of each part.

• A teams best hitter has a life time batting average of .320. He has been at bat 325 times. – how many hits has he made?

4

workout 1) 3x + 2x = 75x = 7 --> 7/53(7/5) = 21/5 meters2(7/5) = 14/5 meters

2) x/325 = 32/100100x = 325 x 32100x = 10400 x = 104

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7.2 Properties of Proportions

• Proportion: equation stating that 2 ratios are equal

• Properties (given a/b = c/d) :– b/a = d/c – ad = bc– a/c = b/d– a+b/b = c+d/d examples: (given a/b = 3/5)

1. 5a = 3b2. 5/b = 3/a3. a+b/b = 3+5/5 --> 8/54. 5/3 = b/a

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Proportion Practice Problems

• Choose yes or no– given: 10/20 = a/b

– is 10 x b = 20 x a ?– is 10/20 = b/a ?– is 30/20 = a+b/b ?– is 20/10 = b/a ?– 10/a = b/20?

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ANSWERS YES b.c ad = bc

NO b.c a/b no= d/c

YES b.c a+b/b = c+d/d

YES b.c b/a = d/c

NO b.c a/c no= d/b

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7.3 Similar Polygons• recall congruent triangles

– corresponding angles --> congruent– corresponding sides --> congruent

Similar triangles

A

B

C D

E

F

-Corresponding angles are congruent -Corresponding sides are in proportion-AB/DE = BC/EF = AC/DF

9

Examples: find length of EF if triABC is similar to triDEF

A

B

C

D

E

F

2

3

4 4

6

x

2/4 = 4/x 2x = 16x = 8

7-4 A Postulate for Similar Triangles

• Postulate 15: AA Similarity Postulate- If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Example: Are these triangle similar? How?

Conclude: yes, AA Similarity (AA~)

CA

B

Practice Problemso Determine if the triangles are similar and how.1)

2) Given: Both Triangles are Isosceles

50

40

555 5

Answers

• 1. 40 + 90 + x = 180 x = 5050 + 90 + x = 180 x = 40

they are similar by AA similarity

7-5 Theorems for Similar Triangles

• Theorem 7-1 (SAS Similarity Theorem)- If an angle of one triangle is congruent to an angle of another

triangle and the sides including those angles are in proportion, then the triangles are similar.

Example: Are these Triangles congruent?Why?Answer: Yes, SAS~

1x y

E F

A

B C

Given: Angle A is congruent to Angle D AB/DE = AC/DF

7-5 Continued

• Theorem 7-2 (SSS Similarity Theorem)- If the sides of two triangles are in proportion, then the

triangles are similar. Example:

Answer: Yes, SSS~ A

BC

D

E F

1X Y

Given: AB/DE = BC/EF = AC/DF

Practice Problems

• 1.

• 2.

6

9

A

B

E

D

C

10

15

7.5

65K 12 M

LP

N

O

658

5

Answers

• 1. Triangle BAC ~ Triangle EDC; SAS~• 2. Triangle LKM ~ Triangle NPO; SAS~

7-6 Proportional Lengths

• Theorem 7-3 (Triangle Proportionality Theorem)- If a line parallel to one side of a triangle intersects the other

two sides, then it divides those sides proportionally. Example:

Find the numerical value ofA) TN/ NRB) TR/NRC) RN/RT M N

R

S T

6

3Answer: a) tn/nr = sm/mr = 3/6 = ½b) Tr/nr = sr/mr = 9/6 = 3/2 c) Rn/rt = rm/rs = 6/9 = 2/3

7-6 Continued

• Corollary- If three parallel lines intersect two

transversals, then they divide the transversals proportionally.

• Theorem 7-4 (Triangle Bisector Theorem)- If a ray bisects an angle of a triangle, then it

divides the opposite side into segments proportional to the other two sides.

7-6 Continued

Example:

12

F

E

G

D

K

3

4Given: Triangle DEF; Ray DG bisects Angle FDEProve: GF/ GE = DF/DE

Answer: (Plan for Proof)Draw a line through E parallel to Ray DGAnd intersecting Ray FD at K.Apply Triangle Proportionality Theorem To Triangle FKE. Triangle DEK is isosceles With DK = DE. Substitute this into your Proportion to complete the proof.

Practice Problems

1. State a proportion for the diagram:

an

g

b

Answer

• 1. a/n = b/g

Practice Proof

• Given: Angle H and Angle F are right triangles Prove: HK * GO = FG * KO

Statements Reasons 1

2O

K

H

F G

Answer to Proof

Statements Reasons

1. Angle 1 is congruent to Angle 2.

2. Angle H and Angle F are right Triangles.

3. Angle H = 90 and Angle F = 90

4. Angle H is congruent to Angle F.

5. Triangle HKO ~ Triangle FGO

6. HK/FG = KO/GO7. HK*GO = FG*KO

1. Vertical Triangles are congruent.

2. Given3. Def. of right

triangle.4. Def. of congruent

triangle5. AA~6. Corr. Sides of ~

Triangles are in proportion.

7. A property of proportions.