6.4 and 6.5 Congruent and Similar Triangles

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# 6.4 and 6.5 Congruent and Similar Triangles - PowerPoint PPT Presentation

6.4 and 6.5 Congruent and Similar Triangles. Similar and Congruent Figures. 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. Tests for Congruency. Ways to prove triangles congruent :

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Presentation Transcript
Similar and Congruent Figures
• 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.
Tests for Congruency

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 )
Thinking Time ?????
• 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

Similar triangles

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

Similar triangles are like similar polygons. Their corresponding angles are CONGRUENT and their corresponding sides are PROPORTIONAL.

10

5

6

3

8

4

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

AA Similarity
• 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

SSS Similarity
• 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

SSS Similarity Theorem

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

D

A

C

B

F

E

SAS Similarity
• 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

SAS Similarity Theorem

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