Proportions in Similar Triangles

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Proportions in Similar Triangles
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• GSP c8\sss_sas.gsp

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SSS Theorem

• If corresponding sides of two triangles are
  proportional then the two triangles are similar.

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SAS Theorem

• If two corresponding sides of two triangles are
  proportional and the included angles are
  congruent then the two triangles are similar.

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Side Splitter Theorem

• If a line is parallel to one side of a triangle
  and intersects the other two sides, it divides
  those two sides proportionally.

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Side to Side Problem
Side Side
Side Side
C
gtgt
B
D
gtgt
E
A
gtgt
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• Theorem If three or more parallel lines are
  intersected by tow transversals, the parallel
  lines divide the transversals proportionally.

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Side to Base Problem
?CBD?CAE
Side Base
Side Base
Side Base
Side Side
C
Base Base
B
D
gtgt
E
A
gtgt
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Corollary

• If three or more parallel lines intersect two
  transversals, then they cut off the transversals
  proportionally.
• Hint Many proportions are created in this
  diagram... just a few are provided.  With any
  proportion simply make sure that corresponding
  parts line up.

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Corollary

• If three or more parallel lines cut off congruent
  segments on one transversal, then they cut off
  congruent segments on every transversal.

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Side to Side Problem
Side Side Side
Side Side Side
A
E
gtgt
B
F
gtgt
G
C
gtgt
D
H
gtgt
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Triangle Midpoint Proportionality Theorem

• A segment whose endpoints are the midpoints of
  two sides of a triangle is parallel to the third
  side of the triangle, and its length is one-half
  the length of the third side. Hint CBCA 12,
  CDCE 12 and BDAE 12 

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Converse of the Triangle Proportionality Theorem

• If a line intersects two sides of a triangle and
  separates the sides into corresponding segments
  of proportional lengths, then the line is
  parallel to the third side.
• If a line intersects two sides of a triangle and
  separates the sides into corresponding segments
  of proportional lengths, then the line is
  parallel to the third side. 

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Triangle Proportionality Theorem

• If a line is parallel to one side of a triangle
  and intersects the other two sides in two
  distinct points, then it separates these sides
  into segments of proportional lengths.

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Angle Bisector Theorem

• If a ray bisects an angle of a triangle, it
  divides the opposite side into segments that are
  proportional to the adjacent sides.

proportional angle segments.gsp
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Angle Bisector Problem
Side Side
base base
B
//
//
A
D
C
E
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Angle Bisector

• An angle bisector divides the opposite side into
  segments proportional to the lengths of the
  adjacent sides.

B
\
A
\
C
D
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Proportional Perimeters Theorem

• If two triangles are similar, then the perimeters
  are proportional to the measures of corresponding
  sides.

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Proportional Altitudes Theorem

• If two triangles are similar, then the measures
  of the corresponding altitudes are proportional
  to the measures of the corresponding sides.

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Proportional Angle Bisectors Theorem

• If two triangles are similar, then the measures
  of the corresponding angle bisectors are
  proportional to the measures of the corresponding
  sides.

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Proportional Medians Theorem

• If two triangles are similar, then the measures
  of the corresponding medians are proportional to
  the measures of the corresponding sides.

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Practice Problems

• 1.8.6  Find the value of x. 
• A. 6.4 B. 10  C. 5.8 D. 7.6 

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B
B
B
D
D
C
A
?ABC? ?ADB ? ?BDC ? ?
D
C
A
D
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Practice Problems

• 2. 8.6 Find the value of y.   
• A. 1.9 B. 2.2  C. 1.8 D. 2 

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Practice Problems

• 3. 8.6  If a median of a triangle is also an
  angle bisector, then the triangle is_______. 
• A. isosceles
• B. right 
• C. scalene
• D. equilateral

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Practice Problems

• 4. 8.6 The perimeter of HIJ 22, and the
  perimeter of FGH 38. If BH 6, then AH
  _____.   
• 114/11
• B. 66/9 
• C.22
• D.10

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Practice Problems

• 5. 8.6   ?ABC ?XYZ. If BM 5, YZ 4, and XY
  8, then AB ______.
• A.17
• B.10 
• C.20
• D.40 

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Example

• 1.8.4  Find the value of x.   
• A. 9
• B. 7 
• C. 50/7
• D. 8 

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Example

• 2.8.4 In the figure, . If AB 4, DE x 1, BC
  4, and EF 3x - 9, what is the value of x?   
• A. 6
• B. 3 
• C. 5
• D. 4  

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Example

• 3.8.4  Based on the figure below, which statement
  is false?   
• ltLPQ ltLMN
• PQMN
• ?LMN ? LQP
• mltM lt mltN 

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Example

• 4.  If PQBC and PM MQ, which statement is not
  necessarily true?   
• ?APQ is isosceles. 
• ?ABC is isosceles. 
• ?ABC is equilateral. 
• D. ?ABC ?APQ

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Example

• 5.8.4.  If AB 1, WX 2, BD 7, and YZ 3,
  then what is the measure of XY?   
• A.14
• B.11 
• C.4
• D.10

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8.4.7
?ABC ?DEF
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5.5
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3
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