Lesson 5-3: Proving Triangles Similar 1 Lesson 6-3 Similar Triangles The following must occur for triangles to be similar, but there are other short cuts.

Lesson 5-3: Proving Triangles Similar 1

Lesson 6-3Similar Triangles

The following must occur for triangles to be similar, but there are other short cuts to prove if triangles are similar (AA, SSS, SAS) :

1) The angles must be congruent 2) Sides must be proportional

Lesson 5-3: Proving Triangles Similar 2

AA Similarity (Angle-Angle)If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar.

E

DA

B

C F

Conclusion:

andGiven:

Lesson 5-3: Proving Triangles Similar 3

SSS Similarity (Side-Side-Side)If the measures of the corresponding sides of 2 triangles are proportional, then the triangles are similar.E

DA

B

C FGiven:

Conclusion:

5

11 22

8 1610

Lesson 5-3: Proving Triangles Similar 4

SAS Similarity (Side-Angle-Side)If the measures of 2 sides of a triangle are proportional to the measures of 2 corresponding sides of another triangle and the angles between them are congruent, then the triangles are similar.

Given:

Conclusion:

E

DA

B

C F

5

11 22

10

Lesson 5-3: Proving Triangles Similar 5

Similarity is reflexive, symmetric, and transitive.

1. Mark the Given . . . and what it implies.2. Mark … Shared Angles or Vertical Angles3. Choose a Method. (AA, SSS , SAS)

Think about what you need for the chosen method and be sure to include those parts in the proof.

Steps for proving triangles similar:

Proving Triangles Similar

Lesson 5-3: Proving Triangles Similar 6

Problem #1:

Pr :

Given DE FG

ove DEC FGC

CD

E

G

F

Step 1: Mark the given … and what it impliesStep 2: Mark the vertical anglesStep 3: Choose a method: (AA,SSS,SAS)Step 4: List the Parts in the order of the method with reasonsStep 5: Is there more? Statements Reasons

||

▲ ▲

Lesson 5-3: Proving Triangles Similar 7

Problem #2

Step 1: Mark the given … and what it impliesStep 2: Choose a method: (AA,SSS,SAS)Step 4: List the Parts in the order of the method with reasonsStep 5: Is there more? Statements Reasons

: 3 3 3Pr :Given IJ LN JK NP IK LP

ove IJK LNP

N

L

P

I

J K 2. IJLN=3,

JKNP=3,

IKLP=3

3. IJLN=

JKNP=

IKLP

▲ ▲

Lesson 5-3: Proving Triangles Similar 8

Problem #3

Step 1: Mark the given … and what it implies

Step 3: Choose a method: (AA,SSS,SAS)

: int

intPr :

Given G is the midpo of ED

H is the midpo of EFove EGH EDF

E

D F

G H

Step 2: Mark the reflexive angles

Lesson 5-3: Proving Triangles Similar 9

Determine whether each pair of triangles is similar. Justify your answer (AA, SSS, or SAS Similarity).

3 4

5

4.5 6

7.5

6

8

12

1620

36

18

24

A

B

C

D

E

F

A

B C

ED

A

B

C

D

E

F

A

B C

Lesson 5-3: Proving Triangles Similar 10

Identify the similar triangles, and find x and the measure of the indicated sides.

x + 33

5 2x - 8

A

B

C

D

E

AB and BC

Lesson 5-3: Proving Triangles Similar 11

Identify the similar triangles, and find x and the measure of the indicated sides.

8

5

x + 2

6

A

E B

DC

AB and AC

Lesson 5-3: Proving Triangles Similar 12

If PR || KL, KN = 9, LN = 16, PM = 2(KP), find KP, KM, MR, ML, MN, and PR.

L N K

R Q P

M