Transcript Mathematical Ideas
Chapter 9 Geometry
© 2008 Pearson Addison-Wesley.
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Chapter 9: Geometry
9.1 9.2 9.3 9.4 Points, Lines, Planes, and Angles Curves, Polygons, and Circles Perimeter, Area, and Circumference The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem 9.5
9.6
Space Figures, Volume, and Surface Area Transformational Geometry 9.7 Non-Euclidean Geometry, Topology, and Networks 9.8 Chaos and Fractal Geometry
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Chapter 1
Section 9-4
The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem
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The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem • Congruent Triangles • Similar Triangles • The Pythagorean Theorem
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Congruent Triangles
Triangles that are both the same size and same shape are called
congruent triangles
.
E B A D F
The corresponding sides are congruent and corresponding angles have equal measures. Notation:
ABC
DEF
.
C
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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.
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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.
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Congruence Properties - SSS
Side-Side-Side (SSS)
congruent.
If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are
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Example: Proving Congruence (SAS)
Given
:
Prove: CE = ED AE = EB
ACE
BDE
Proof STATEMENTS REASONS
A C
1.
CE = ED
2.
AE = EB
3.
4.
CEA
DEB
ACE
BDE
1. Given 2. Given 3. Vertical Angles are equal 4. SAS property
E D B
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Example: Proving Congruence (ASA)
Given
:
Prove: ADB ABD
ADB
CBD CDB
CDB
Proof STATEMENTS REASONS
A
1. 2.
ADB
ABD
CBD CDB
1. Given 2. Given 3.
DB = DB
4.
ADB
CDB B
3. Reflexive property 4. ASA property
D
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C
9-4-10
Example: Proving Congruence (SSS)
Given
:
Prove: AD = CD AB = CB
ABD
CDB
Proof STATEMENTS REASONS
A D
1. A
D = CD
2.
AB = CB
3.
BD = BD
4.
ABD
CDB
1. Given 2. Given 3. Reflexive property 4. SSS property
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B C
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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 following properties hold.
B
1. The base angles
A
and
C
are equal.
2. Angles
ABD
and
CBD
are equal.
3. Angles
ADB
and
CDB
angles.
D C
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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.
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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.
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Example: Finding Side Length in Similar Triangles
ABC
is similar to
DEF
.
Find the length of side
DF
.
E
8
F
24
D
Solution Set up a proportion with corresponding sides:
EF
DF BC AC
8 16
DF
32
A
Solving, we find that
DF
= 16.
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B
16 32
C
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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
b
2
c
2 .
That is, the sum of the squares of the lengths of the legs is equal to the square of the hypotenuse.
hypotenuse
c
leg
a
leg
b
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Example: Using the Pythagorean Theorem
Find the length
a
in the right triangle below.
a
Solution
a
2
b
2
a
2 36 2
c
2 39 2
a
2
a
2
a
225 15
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36 39
9-4-17
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
b
2
c
2 , then the triangle is a right triangle.
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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 2 7 2 ?
8 2 ?
64 65 64 No, it is not a right triangle.
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