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7-3 Triangle Similarity. I CAN -Use the triangle similarity theorems to determine if two triangles are similar. Use proportions in similar triangles to solve for missing sides. Recall in 7-2, to prove that two polygons are similar you had to:. Show all corresponding angles are congruent.

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7 3 triangle similarity
7-3 Triangle Similarity

  • I CAN

  • -Use the triangle similarity theorems to

  • determine if two triangles are similar.

  • Use proportions in similar triangles to solve

  • for missing sides.


Recall in 7-2, to prove that two polygons are similar you had to:

Show all corresponding angles are congruent

AND

Show all corresponding sides are proportional

Triangle Similarity Theorems are “shortcuts”

for showing two triangles are similar.


Example had to:

D

9

E

B

12

10

5

C

A

6

18

F

Similarity Statement Reason

ABC~

EFD

by AA

Because A E and C D

justification


Example had to:

Explain why the triangles

are similar and write a

similarity statement.

By the Triangle Sum Theorem, mC = 47°, so C F.

B E by the Right Angle Congruence Theorem.

Therefore, ∆ABC ~ ∆DEF by AA ~.


Example had to:

Verify that the triangles are similar.

∆PQR and ∆STU

Therefore ∆PQR ~ ∆STU by SSS ~.


Example had to:

Verify that the triangles are similar.

∆DEF and ∆HJK

D  H by the Definition of Congruent Angles.

Therefore ∆DEF ~ ∆HJK by SAS ~.


Example had to:

Verify that ∆TXU ~ ∆VXW.

TXU VXW by the Vertical Angles Theorem.

Therefore ∆TXU ~ ∆VXW by SAS ~.


Your Turn had to:


Your Turn had to:


Your Turn had to:


Find the value of had to:x such that

∆ACE ~ ∆BCD

C

Why is ∆ACE ~ ∆BCD?

C

3

3

12

D

B

D

B

x

12

28

E

C

A

3 = 12

x + 3 28

x + 3

E

A

12(x + 3) = 84

28

12x + 36 = 84

– 36 – 36

12x = 48

x = 4


Example had to:

Explain why ∆ABE ~ ∆ACD, and then find CD.

Step 1 Prove triangles are similar.

A A by Reflexive Property of , and B C since they are both right angles.

Therefore ∆ABE ~ ∆ACD by AA ~.

Step 2 Find CD.

x(9) = 5(3 + 9)

9x = 60


Example had to:

Explain why ∆RSV ~ ∆RTU and then find RT.

Step 1 Prove triangles are similar.

It is given that S T.

R R by Reflexive Property of .

Therefore ∆RSV ~ ∆RTU by AA ~.

Step 2 Find RT.

RT(8) = 10(12)

8RT = 120

RT = 15


Your Turn had to:


Properties of similar triangles 7 4

Properties of Similar Triangles had to:7-4

  • I can use the triangle proportionality theorem and its converse.

  • I can set up and solve problems using properties of similar triangles.



It is given that , so by the Triangle Proportionality Theorem.

Example

Find US.

US(10) = 56


Example Triangle Proportionality Theorem.

Find PN.

2PN = 15

PN = 7.5


Your Turn Triangle Proportionality Theorem.

Solve for x.


Example Triangle Proportionality Theorem.

Solve for x.


Example Triangle Proportionality Theorem.


Example Triangle Proportionality Theorem.

I

Solve for x, y, and w.

x

4

N

E

w

y

5

L

A

3

2

D

S

12


Example Triangle Proportionality Theorem.

I

Solve for x, y, and w.

x = 6

4

N

E

w

y

5

L

A

3

2

D

S

12

OR

OR


Example Triangle Proportionality Theorem.

I

Solve for x, y, and w.

x = 6

4

N

E

w

5

L

A

3

2

D

S

12


Example Triangle Proportionality Theorem.

BD = 8; DF = 6; CE = 16.

EG = ________


Example Triangle Proportionality Theorem.

BD = 2x – 2; DF = 4; CE = x + 2; EG = 8

Find BD and CE


Your Turn Triangle Proportionality Theorem.


Example: SHOW EF Triangle Proportionality Theorem. BC if BE = 21,

AE = 42, CF = 15, and AF = 30

?

So, EF BC


Verify that . Triangle Proportionality Theorem.

Since , by the Converse of the Triangle Proportionality Theorem.

Example


AC Triangle Proportionality Theorem. = 36 cm, and BC = 27 cm. Verify that .

Since , by the Converse of the

Triangle Proportionality Theorem.

Your Turn



by the conclusion.∆ Bisector Theorem.

Example

Find PS and SR.

PS = x – 2

40(x – 2) = 32(x + 5)

= 30 – 2 = 28

40x – 80 = 32x + 160

SR = x + 5

40x – 80 = 32x + 160

= 30 + 5 = 35

8x = 240

x = 30


by the conclusion.∆ Bisector Theorem.

Example

Find AC and DC.

4y = 4.5y – 9

–0.5y = –9

y = 18

So DC = 9 and AC = 16.


Your Turn conclusion.


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