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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. Given: Conclusion:
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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