Similarity

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					6.4-6.5: Similarity Shortcuts
                 Objectives:
1. To find missing measures in similar
   polygons
2. To discover shortcuts for determining that
   two triangles are similar
Similar Polygons
Two polygons are similar polygons iff the
 corresponding angles are congruent and the
 corresponding sides are proportional.
                                                    N
   Similarity Statement:            C                               M
       CORN ~ MAIZ
  Corresponding Angles:         O
   C  M     O  N
                                                R
         C
                     A                      M               A
                                                                Z


    R  I    N  Z
StatementOof Proportionality:
                  R
    CO  OR  RN  NC                   A
    MA AI IZ ZM                                         I
Example 1
Triangles ABC
  and ADE are         D
                                     B
  similar. Find the
                                                        A
  value of x.
                          9 cm       6 cm
                                                 8 cm

                                             C
                                         x
                                 E
Example 2
Are the triangles below similar?

                                                  8

             4                     6
 3
                 37

                                  53        10
         5




     Do you really have to check all the sides and angles?
Investigation 1
In this Investigation we will check the first
  similarity shortcut. If the angles in two
  triangles are congruent, are the triangles
  necessarily similar?
                                   F

          C




    50        40   B       50          40
A                        D                      E
Investigation 1
Step 1: Draw ΔABC where m<A and m<B
  equal sensible values of your choosing.



          C




    50       40   B
A
Investigation 1
Step 1: Draw ΔABC where m<A and m<B
  equal sensible values of your choosing.
Step 2: Draw ΔDEF where m<D = m<A and
  m<E = m<B and AB ≠ DE.
                                  F

          C




    50       40   B       50       40
A                       D                   E
Investigation 1
Now, are your triangles similar? What would
 you have to check to determine if they are
 similar?

                                  F

          C




    50       40   B       50       40
A                       D                   E
Angle-Angle Similarity
AA Similarity
   Postulate
If two angles of one
   triangle are
   congruent to two
   angles of another
   triangle, then the two
   triangles are similar.
Example 3
Determine whether the triangles are similar.
 Write a similarity statement for each set of
 similar figures.
Thales
The Greek mathematician
 Thales was the first to
 measure the height of a
 pyramid by using
 geometry. He showed
 that the ratio of a
 pyramid to a staff was
 equal to the ratio of one
 shadow to another.
Example 4
If the shadow of the pyramid is 576 feet, the
   shadow of the staff is 6 feet, and the height
   of the staff is 5 feet, find the height of the
   pyramid.
Example 5
Explain why Thales’ method worked to find
 the height of the pyramid?
Example 6
If a person 5 feet tall casts a 6-foot shadow
   at the same time that a lamppost casts an
   18-foot shadow, what is the height of the
   lamppost?
Investigation 2
What if you decide to
 indirectly measure a
 height on a day when
 there are no shadows?
 The following GSP
 Animation will help you
 discover an alternate
 method of indirect
 measurement using a
 mirror.
Example 7
Your eye is 168
 centimeters from
 the ground and you
 are 114 centimeters
 from the mirror.
 The mirror is 570
 centimeters from
 the flagpole. How
 tall is the flagpole?
Investigation 3
Each group will be given one
 of the three candidates for
 similarity shortcuts. Each
 group member should start
 with a different triangle and
 complete the steps outlined
 for the investigation.
 Share your results and
 make a conjecture based
 on your findings.
Side-Side-Side Similarity
SSS Similarity Theorem:
If the corresponding side lengths of two
   triangles are proportional, then the two
   triangles are similar.
Side-Angle-Side Similarity
SAS Similarity Theorem:
If two sides of one triangle are proportional to two
   sides of another triangle and the included angles
   are congruent, then the two triangles are similar.
Example 8
Are the triangles below similar? Why or why
  not?
Example 9
Use your new conjectures to find the missing
 measure.
                        28
              24



                                   24
                    x

                                        18
                               y
Example 10
Find the value of x that makes ΔABC ~
  ΔDEF.
Assignment
• P. 384-387: 1-4, 7,
  8, 10, 12, 14-17,
  20, 30, 31, 32, 36,
  41, 42
• P. 391-395: 4, 6-8,
  10-14, 33, 39, 40
• Challenge
  Problems

				
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posted:8/16/2011
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