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7-3 Triangle SimilarityPowerPoint Presentation

7-3 Triangle Similarity

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### Properties of Similar Triangles had to:7-4

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, mC = 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:

- I can use the triangle proportionality theorem and its converse.
- I can set up and solve problems using properties of similar triangles.

Example: AF = 3, FC = 5, AE = 2. Find BE had to:

OR

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 = 6

4

N

E

w

y

5

L

A

3

2

D

S

12

OR

OR

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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