Ch 7

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Ch 7

• Similarity

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Lesson 7-1

• Using Proportions

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Ratio
A ratio is a comparison of two numbers such as a
b.
Ratio
When writing a ratio, always express it in
simplest form.
Example
Now try to reduce the fraction.
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Example.
A baseball player goes to bat 348 times and gets
107 hits. What is the players batting average?
Solution
Set up a ratio that compares the number of hits
to the number of times he goes to bat.
Convert this fraction to a decimal rounded to
three decimal places.
The baseball players batting average is 0.307
which means he is getting approxiamately one hit
every three times at bat.
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Proportion
Proportion An equation that states that two
ratios are equal.
Terms
First Term
Third Term
Second Term
Fourth Term
To solve a proportion, cross multiply the
proportion
Extremes a and d
Means b and c
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Proportions- examples.
Example 1
Find the value of x.
Solve the proportion.
Example 2
8x 30
x 3.75
8x 30 8 8
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Lesson 7-2

• Similar Polygons

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Similar Polygons
Definition Two polygons are similar if 1.
Corresponding angles are congruent. 2.
Corresponding sides are in proportion.
Two polygons are similar if they have the same
shape not necessarily have the same size.
The scale factor is the ratio between a pair of
corresponding sides.
Scale Factor
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Naming Similar Polygons
When naming similar polygons, the vertices
(angles, sides) must be named in the
corresponding order.
P
Q
A
B
C
D
S
R
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Example-
The two polygons are similar. Solve for x, y and
z.
Step1 Write the proportion of the sides.
Step 2 Replace the proportion with values.
Step 3 Find the scale factor between the two
polygons.
Note The scale factor has the larger
quadrilateral in the numerator and the smaller
quadrilateral in the denominator.
Step 4 Write separate proportions for each
missing side and solve.
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Example
If ?ABC ?ZYX, find the scale factor from ?ABC
to ?ZYX.
Scale factor is same as the ratio of the sides.
Always put the first polygon mentioned in the
numerator.
The scale factor from ?ABC to ?ZYX is 2/1.
½
What is the scale factor from ?ZYX to ?ABC?
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Lesson 7-3
Proving Triangles Similar
(AA, SSS, SAS)
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AA Similarity (Angle-Angle)
If 2 angles of one triangle are congruent to 2
angles of another triangle, then the triangles
are similar.
and
Given
Conclusion
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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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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
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Similarity is reflexive, symmetric, and
transitive.
Proving Triangles Similar
Steps for proving triangles similar
1. Mark the Given. 2. Mark Shared Angles or
Vertical Angles 3. 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.
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Problem 1
Step 1 Mark the given and what it implies
Step 2 Mark the vertical angles
AA
Step 3 Choose a method (AA,SSS,SAS)
Step 4 List the Parts in the order of the method
with reasons
Step 5 Is there more?
Statements Reasons




Given
Alternate Interior lts
Alternate Interior lts
AA Similarity
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Problem 2
Step 1 Mark the given and what it implies
SSS
Step 2 Choose a method (AA,SSS,SAS)
Step 4 List the Parts in the order of the method
with reasons
Step 5 Is there more?
Statements Reasons




Given
1. IJ 3LN JK 3NP IK 3LP
Division Property
Substitution
SSS Similarity
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Problem 3
Step 1 Mark the given and what it implies
Step 2 Mark the reflexive angles
SAS
Step 3 Choose a method (AA,SSS,SAS)
Step 4 List the Parts in the order of the method
with reasons Next Slide.
Step 5 Is there more?
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Statements Reasons
G is the Midpoint of H is the Midpoint of Given
2. EG DG and EH HF Def. of Midpoint
3. ED EG GD and EF EH HF Segment Addition Post.
4. ED 2 EG and EF 2 EH Substitution
Division Property
Substitution
Reflexive Property
SAS Postulate
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Proportional Parts
Lesson 7-4
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Two polygons are similar if and only if their
corresponding angles are congruent and the
measures of their corresponding sides are
proportional.
Similar Polygons
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Side Splitter 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 length.
Converse
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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Examples
If BE 6, EA 4, and BD 9, find DC.
Example 1
6x 36
x 6
Solve for x.
Example 2
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Midsegment Theorem
A segment that joins 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.
R
M
L
T
S
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Extension of Side Splitter
If three or more parallel lines have two
transversals, they cut off the transversals
proportionally.
If three or more parallel lines cut off congruent
segments on one transversal, then they cut off
congruent segments on every transversal.
F
E
D
A
B
C
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Forgotten Theorem
An angle bisector in a triangle separates the
opposite side into segments that have the same
ratio as the other two sides.
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If two triangles are similar
(1) then the perimeters are proportional to the
measures of the corresponding sides.
(2) then the measures of the corresponding
altitudes are proportional to the measure of the
corresponding sides..
(3) then the measures of the corresponding angle
bisectors of the triangles are proportional to
the measures of the corresponding sides..
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Example
Given ?ABC ?DEF, AB 15, AC 20, BC 25,
and DF 4. Find the perimeter of ?DEF.
The perimeter of ?ABC is 15 20 25 60. Side
DF corresponds to side AC, so we can set up a
proportion as