1. Topic 7: Similarity 7-1: Properties of Proportions
7-2: Applications of Proportions
7-3: Similar Polygons
7-4: Similar Triangles
7-5: Proportional Parts
7-6: Parts of Similar Triangles
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2. 7-1: Properties of Proportions
Definition:
A ratio is a comparison between two numbers that can be expressed three different ways
If a and b are Real Numbers and b  0,
a to b, a:b, and all mean the same thing
Definition:
Two equal ratios are called a proportion
and
are examples of proportions
Topic 7
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3. 7-1: Properties of Proportions
Equality of Cross Products:
For any numbers a and c and any nonzero numbers b and d,
if and only if ad = bc.
In the proportion notice that
Example: a c d b =
a and d are called the extremes
b and c are called the means
Given:
In a proportion the product of the extremes equals the product of the means.
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Section 7-2

4. 7-2: Applications of Proportions
Definition:
A rate is a ratio that compares two different types of units
If you travel 240 miles in 4 hours your average rate is 60 miles per hour
Example:
Expressed as a Proportion:
Notice
In the Proportion
Topic 7
Section 7-3

5. 7-3: Similar Polygons Definition: Topic 7 Section 7-4
Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.
Usually similar objects have the same shape but are different sizes
If ABCD is similar to WXYZ then
Sides are Proportional
Scale Factor
The ratio of corresponding sides is called the scale factor.
Topic 7
Section 7-4

6. 7-4:Similar Triangles Angle-Angle (AA) Postulate:
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.  
If A  A, and AED  ACB, then *ACB  AED
* The symbol  is used to indicate similarity
SSS Similarity Theorem:
If the measures of corresponding sides of two triangles are proportional, then the triangles are similar
If then
ACB  AED
Topic 7
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7. 7-4: Similar Triangles SAS Similarity Theorem:
If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.
If and B  Y, then ABC  XYZ
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8. 7-4: Similar Triangles Theorem:
Similarity of Triangles is Reflexive, Symmetric and Transitive.
If ABC  DEF, and DEF  GHI, then ABC  GHI
If ABC  DEF, then DEF  ABC
ABC  ABC
Transitive Property
Symmetric Property
Reflexive Property
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Section 7-5

9. 7-5: Proportional Parts 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.
If DE ll CB, then
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 , then DE ll CB
Topic 7
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10. 7-5: Proportional Parts 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.
If W is the midpoint of VX and Y is the midpoint of XZ, then WY ll VZ and
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11. 7-5: Proportional Parts Corollary: If p ll q ll r then, AND Back Next
If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally
If p ll q ll r then,
AND
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12. 7-5: Proportional Parts Corollary:
If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal
If p ll q ll r and AB  BC, then XY  YZ.
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Section 7-6

13. 7-6: Parts of Similar Triangles
Proportional Perimeter Theorem:
If two triangles are similar, then their perimeters are proportional to the measures of corresponding sides.
Let P1 = a + b + c, the perimeter of ABC
Let P2 = d + e + f, the perimeter of DEF
If ABC  DEF, then
Topic 7
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14. 7-6: Parts of Similar Triangles
Theorem:
If two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the corresponding sides
If ABC  DEF and CP and FQ are corresponding altitudes, then
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15. 7-6: Parts of Similar Triangles
Theorem:
If two triangles are similar, then the measures of the corresponding angle bisectors are proportional to the measures of the corresponding sides
If PQR  XYZ and QG and YH are corresponding angle bisectors, then
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16. 7-6: Parts of Similar Triangles
Theorem:
If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides
If PQR  XYZ and RF and ZG are corresponding medians, then
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17. 7-6: Parts of Similar Triangles
Theorem:
An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.
Given ABC with 1   2 2 1
Given ABC with CD the angle bisector of ACB,
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Right Triangles & Trigonometry