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6.4 and 6.5 Congruent and Similar Triangles

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- Congruent polygons have all sides congruent and all angles congruent.
- Similar polygons have the same shape; they may or may not have the same size.

Ways to prove triangles congruent :

- SSS ( Side – Side – Side )
- SAS ( Side – Angle – Side )
- ASA ( Angle – Side – Angle ) or AAS ( Angle –Angle – Side )
- RHS ( Right angle – Hypotenuse – Side )

- If 3 angles on A are equal to the 3 corresponding angles on the other B, are the two triangles congruent ?

?

2cm

4 cm

65o

25o

?

12cm

For two similar triangles,

- Similar triangles are triangles with the same shape

- corresponding angles have the same measure
- length of corresponding sides have the same ratio

Example

Side = 6 cm

Angle = 90o

Similar Triangles

3 Ways to Prove Triangles Similar

10

5

6

3

8

4

But you don’t need ALL that information to be able to tell that two triangles are similar….

- If two angles of a triangle are congruent to the two corresponding angles of another triangle, then the triangles are similar.

25 degrees

25 degrees

- If all three sides of a triangle are proportional to the corresponding sides of another triangle, then the two triangles are similar.

21

14

18

8

12

12

If the sides of two triangles are in proportion, then the triangles are similar.

D

A

C

B

F

E

- If two sides of a triangle are proportional to two corresponding sides of another triangle AND the angles between those sides are congruent, then the triangles are similar.

14

21

18

12

D

A

C

B

F

E

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.

D

A

C

B

F

E

SAS Similarity Theorem

Idea for proof

A

80

D

E

80

B

C

ABC ~ ADE by AA ~ Postulate

C

6

10

D

E

5

3

A

B

CDE~ CAB by SAS ~ Theorem

L

5

3

M

6

6

K

N

6

10

O

KLM~ KON by SSS ~ Theorem

A

20

D

30

24

16

B

C

36

ACB~ DCA by SSS ~ Theorem

L

15

P

A

25

9

N

LNP~ ANL by SAS ~ Theorem