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Geometry
Chapter 8 Notes
8.1
Ratio & Proportion
If a and b are two quantities that are measured in the sane units, then the ratio of a to b is
written
a or a:b
b
Ex: If a soccer team won 102 games and lost 2, the ratio of wins would be 12 or 6:1
2
Cross Product Property
The product of the extremes equals the product of the means
If a = c then ad = bc
b d
Ex: Solve x = 8
3
2
2x = 24, x = 12
Reciprocal Property
If two ratios are equal, then their reciprocals are also equal.
If a = c then b = d
b d
a c
Ex: If
8.2
4 = 5
x
7
then
x = 7
4
5
Problem Solving in Geometry with Proportions
Additional Properties of Proportions
If a = c then a = b
b d
c d
Ex: If
x = y then
6
10
If a = c then
b d
Ex: If
x = y
3
4
x = 6
y 10
a+b = c+d
b
d
then
x+3 = y+4
3
4
8.3
Similar Polygons
When there is a correspondence between two polygons such that their corresponding
angles are congruent and the lengths of corresponding sides are proportional the two
polygons are called similar polygons.
In the diagram, ABCD is similar to EFGH.
The symbol ∼ is used to indicate similarity.
So, ABCD ∼ EFGH.
Theorem 8.1
Example: List all the pairs of congruent angles and write the statement of
proportionality for the figures where ABC ∼ DEF
A  D
B  E
C  F
E
B
A
C
D
F
Statement of proportionality
AB = BC = CA
DE EF
FD
8.4
Similar Triangles
One method of proving triangles are similar is the use of the AA similarity theorem.
Angle-Angle (AA) Similarity Postulate - Postulate 25
B
Example: Write a two-column proof:
Given DE is a midsegment of ABC
Prove: ABC ∼ DBE
1.
2.
3.
4.
5.
DE is Midsgement of ABC
DE ǀǀ AC
CAB   EDB
 B  B
ABC ∼ DBE
D
E
A
C
1. Given
2. Midsegment Theorem
3.Corresponding Angles Postulate
4. Reflexsive Property of Congruence
5. AA Similarity Postulate
8.5 Proving Triangles are Similar
Additional methods of proving triangles are similar involve the SSS & SAS similarity
theorems.
Side-Side-Side (SSS) Similarity Theorem (Theorem 8.2)
Side-Angle-Side (SAS) Similarity Theorem (Theorem 8.3)
Example: Write a two-column proof:
Given: ABC is equilateral, and DE, DF, EF are midsegments
Prove: ABC ∼ FED
1.
2.
3.
4.
5.
ABC is equilateral
DE, DF, EF are midsegments
AB = BC = AC
DE = ½BC,EF = ½AC, DF = ½AB
ABC ∼ FED
1. Given
2. Given
3. Definition of equilateral 
4. Midsegment Theorem
5. SSS Similarity Postulate
B
D
E
A
C
F
8.6
Proportions and Similar Triangles
There are four proportionality theorems that use similar triangles to prove each
theorem.
Triangle Proportinality Theorem (Theorem 8.4)
Converse of the Triangle ProportionalityTheorem (Theorem 8.5)
Paraellel Lines and Transversals Theorem (Theorem 8.6)
Ray bisector and Opposite Side Theorem (Theorem 8.7)
Example: Write a two-column proof:
Given: GB ǀǀ FC ǀǀ ED
Prove: ABG ∼ ADE
1.
2.
3.
4.
GB ǀǀ FC ǀǀ ED
ABG   ADE
GAB   EAD
ABG ∼ ADE
A
1. Given
2. Corresponding ’s Post.
3. Reflexsive Prop. of 
4. AA Similarity Postulate
D
F
E
B
C
D
8.7
Dilations