Chapter 9 Geometry

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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

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 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

9-4-15

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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