geometry of triangles

Chapter 9

Geometry

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Chapter 9: Geometry

9.1 Points, Lines, Planes, and Angles 9.2 Curves, Polygons, and Circles9.3 Perimeter, Area, and Circumference 9.4 The Geometry of Triangles: Congruence, Similarity,

and the Pythagorean Theorem9.5 Space Figures, Volume, and Surface Area9.6 Transformational Geometry9.7 Non-Euclidean Geometry, Topology, and Networks9.8 Chaos and Fractal Geometry

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Chapter 1

Section 9-4The 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

A

B

C

Triangles that are both the same size and same shape are called congruent triangles.

D

E

FThe corresponding sides are congruent and corresponding angles have equal measures. Notation: .ABC DEF

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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) If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent.

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Example: Proving Congruence (SAS)

Given: CE = EDAE = EB

Prove: ACE BDE

STATEMENTS REASONS

1. CE = ED 1. Given

2. AE = EB 2. Given

3. 3. Vertical Angles are equal

4. 4. SAS property

A

B

DE

C

ACE BDE

Proof

CEA DEB

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Example: Proving Congruence (ASA)

Given:

Prove: ADB CDB

STATEMENTS REASONS

1. 1. Given

2. 2. Given

3. DB = DB 3. Reflexive property

4. 4. ASA property

A

B

D

C

Proof

ADB CBD ABD CDB

ADB CDB

ADB CBD ABD CDB

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Example: Proving Congruence (SSS)

Given: AD = CDAB = CB

Prove: ABD CDB

STATEMENTS REASONS

1. AD = CD 1. Given

2. AB = CB 2. Given

3. BD = BD 3. Reflexive property

4. 4. SSS property

A

B

D C

ABD CDB

Proof

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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.

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 C

B

D

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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

is similar to .ABC DEF Find the length of side DF.

A

B

CD

EF

Solution

1624

32

8

Set up a proportion with corresponding sides:

EF DFBC AC

816 32

DF Solving, we find that DF = 16.

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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

That is, the sum of the squares of the lengths of the legs is equal to the square of the hypotenuse.

2 2 2.a b c

leg a

leg b

hypotenuse c

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Example: Using the Pythagorean Theorem

Find the length a in the right triangle below.

Solution2 2 2a b c

39

36

a

2 2 236 39a 2 1296 1521a

2 225a 15a

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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

then the triangle is a right triangle.2 2 2 ,a b c

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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?

Solution2 2 24 7 8

?

16 49 64 ?

65 64No, it is not a right triangle.