Download Mathematical Ideas

Chapter 9
Geometry
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Chapter 9: Geometry
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
Points, Lines, Planes, and Angles
Curves, Polygons, and Circles
Perimeter, Area, and Circumference
The Geometry of Triangles: Congruence,
Similarity, and the Pythagorean Theorem
Space Figures, Volume, and Surface Area
Transformational Geometry
Non-Euclidean Geometry, Topology, and Networks
Chaos and Fractal Geometry
9-4-2
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Chapter 1
Section 9-4
The Geometry of Triangles:
Congruence, Similarity, and the
Pythagorean Theorem
9-4-3
© 2008 Pearson Addison-Wesley. All rights reserved
The Geometry of Triangles: Congruence,
Similarity, and the Pythagorean Theorem
• Congruent Triangles
• Similar Triangles
• The Pythagorean Theorem
9-4-4
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Congruent Triangles
Triangles that are both the same size and same
shape are called congruent triangles.
B
E
A
D
F
C
The corresponding sides are congruent and
corresponding angles have equal measures.
Notation: ABC  DEF.
9-4-5
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Congruence Properties - SAS
Side-Angle-Side (SAS) If two sides and the
included angle of one triangle are equal,
respectively, to two sides and the included
angle of a second triangle, then the triangles
are congruent.
9-4-6
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Congruence Properties - ASA
Angle-Side-Angle (ASA) If two angles and
the included side of one triangle are equal,
respectively, to two angles and the included
side of a second triangle, then the triangles
are congruent.
9-4-7
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Congruence Properties - SSS
Side-Side-Side (SSS) If three sides of one
triangle are equal, respectively, to three sides
of a second triangle, then the triangles are
congruent.
9-4-8
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Example: Proving Congruence (SAS)
Given:
CE = ED
AE = EB
C
Prove:
ACE  BDE
A
Proof
STATEMENTS
REASONS
1. CE = ED
1. Given
2. AE = EB
2. Given
3.
B
E
D
CEA  DEB 3. Vertical Angles are equal
4. ACE  BDE
4. SAS property
9-4-9
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Example: Proving Congruence (ASA)
Given:
ADB  CBD
ABD  CDB
ADB  CDB
Prove:
Proof
STATEMENTS
1.
2.
B
A
C
D
REASONS
ADB  CBD 1. Given
ABD  CDB 2. Given
3. DB = DB
3. Reflexive property
4. ADB  CDB
4. ASA property
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9-4-10
Example: Proving Congruence (SSS)
Given:
AD = CD
AB = CB
B
Prove:
ABD  CDB
Proof
A
STATEMENTS
REASONS
D
1. AD = CD
1. Given
2. AB = CB
2. Given
3. BD = BD
3. Reflexive property
4. ABD  CDB
4. SSS property
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C
9-4-11
Important Statements About Isosceles
Triangles
If ∆ABC is an isosceles triangle with AB = CB,
and if D is the midpoint of the base AC, then the
B
following properties hold.
1. The base angles A and C are equal.
2. Angles ABD and CBD are equal.
3. Angles ADB and CDB are both right
angles.
A
D
C
9-4-12
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Similar Triangles
Similar Triangles are pairs of triangles that are
exactly the same shape, but not necessarily the
same size. The following conditions must hold.
1. Corresponding angles must have the same
measure.
2. The ratios of the corresponding sides must be
constant; that is, the corresponding sides are
proportional.
9-4-13
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Angle-Angle (AA) Similarity Property
If the measures of two angles of one triangle
are equal to those of two corresponding
angles of a second triangle, then the two
triangles are similar.
9-4-14
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Example: Finding Side Length in
Similar Triangles
ABC is similar to DEF.
E
8
F
Find the length of side DF.
32
Set up a proportion with
corresponding sides:
EF DF

BC AC
8 DF

16 32
16
24
D
Solution
B
C
A
Solving, we find that DF = 16.
9-4-15
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Pythagorean Theorem
If the two legs of a right triangle have lengths a
and b, and the hypotenuse has length c, then
a 2  b2  c 2 .
That is, the sum of the squares of the lengths of
the legs is equal to the square of the hypotenuse.
leg a
hypotenuse c
leg b
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9-4-16
Example: Using the Pythagorean
Theorem
Find the length a in the right triangle below.
39
a
Solution
a b  c
2
2
2
36
a2  362  392
a2  1296  1521
2
a  225
a  15
9-4-17
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Converse of the Pythagorean Theorem
If the sides of lengths a, b, and c, where c is
the length of the longest side, and if
a 2  b2  c2 ,
then the triangle is a right triangle.
9-4-18
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Example: Applying the Converse of
the Pythagorean Theorem
Is a triangle with sides of length 4, 7, and 8,
a right triangle?
Solution
?
4 7 8
2
2
2
?
16  49  64
65  64
No, it is not a right triangle.
9-4-19
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