Warm-Up

1
Warm-Up

• Since they are polygons, what two things must be
  true about triangles if they are similar?

2
Similar Polygons

• Two polygons are similar polygons iff the
  corresponding angles are congruent and the
  corresponding sides are proportional.

Similarity Statement
Corresponding Angles
Statement of Proportionality
3
Example 1

• Triangles ABC and ADE are similar. Find the
  value of x.

4
Example 2

• Are the triangles below similar?

Do you really have to check all the sides and
angles?
5
6.4-6.5 Similarity Shortcuts

• Objectives
• To find missing measures in similar polygons
• To discover shortcuts for determining that two
  triangles are similar

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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?

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Investigation 1

• Step 1 Draw ?ABC where mltA and mltB equal
  sensible values of your choosing.

8
Investigation 1

• Step 1 Draw ?ABC where mltA and mltB equal
  sensible values of your choosing.
• Step 2 Draw ?DEF where mltD mltA and mltE mltB
  and AB ? DE.

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Investigation 1

• Now, are your triangles similar? What would you
  have to check to determine if they are similar?

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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.

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Example 3

• Determine whether the triangles are similar.
  Write a similarity statement for each set of
  similar figures.

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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.

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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.

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Example 5

• Explain why Thales method worked to find the
  height of the pyramid?

15
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?

16
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.

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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?

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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.

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Side-Side-Side Similarity

• SSS Similarity Theorem
• If the corresponding side lengths of two
  triangles are proportional, then the two
  triangles are similar.

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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.

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Example 8

• Are the triangles below similar? Why or why not?

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Example 9

• Use your new conjectures to find the missing
  measure.

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Example 10

• Find the value of x that makes ?ABC ?DEF.

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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