Download Section 9-4

Chapter 1
Chapter 9
Geometry
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
Triangles that are both the same size and same
shape are called congruent triangles.
B
E
• Congruent Triangles
• Similar Triangles
• The Pythagorean Theorem
A
D
F
C
The corresponding sides are congruent and
corresponding angles have equal measures.
Notation: ∆ABC ≅ ∆DEF .
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Congruence Properties - SAS
Congruence Properties - ASA
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.
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
Example: Proving Congruence (SAS)
Given:
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.
CE = ED
AE = EB
∆ACE ≅ ∆BDE
Prove:
Proof
STATEMENTS
C
B
E
A
D
REASONS
1. CE = ED
1. Given
2. AE = EB
2. Given
3. ∠CEA = ∠BED
3. Vertical Angles are equal
4. ∆ACE ≅ ∆BDE
4. SAS property
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© 2008 Pearson Addison-Wesley. All rights reserved
Example: Proving Congruence (ASA)
Given:
∠ADB = ∠DBC
∠ABD = ∠CDB
∆ADB ≅ ∆CDB
Prove:
Proof
STATEMENTS
B
Given:
C
A
Example: Proving Congruence (SSS)
AD = CD
AB = CB
∆ABD ≅ ∆CDB
Prove:
Proof
STATEMENTS
D
REASONS
B
A
REASONS
D
1. ∠ADB = ∠DBC 1. Given
2. ∠ABD = ∠CDB 2. Given
1. AD = CD
1. Given
2. AB = CB
2. Given
3. DB = DB
3. Reflexive property
3. BD = BD
3. Reflexive property
4. ∆ADB ≅ ∆CDB
4. ASA property
4. ∆ABD ≅ ∆CDB
4. SSS property
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Important Facts About Isosceles
Triangles
Similar 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.
Similar Triangles are pairs of triangles that are
exactly the same shape, but not necessarily the
same size. The following conditions must 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.
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.
A
D
C
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C
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Angle-Angle (AA) Similarity Property
Example: Finding Side Length in
Similar Triangles
E
∆ABC is similar to ∆DEF .
Find the length of side DF.
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.
F
B
16
24
D
Solution
A
Solving, we find that DF = 16.
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Example: Using the Pythagorean
Theorem
Pythagorean Theorem
Find the length a in the right triangle below.
If the two legs of a right triangle have lengths a
and b, and the hypotenuse has length c, then
a 2 + b2 = c2 .
a 2 + b2 = c 2
36
a 2 + 362 = 392
a 2 + 1296 = 1521
a 2 = 225
a = 15
hypotenuse c
leg b
39
a
Solution
That is, the sum of the squares of the lengths of
the legs is equal to the square of the hypotenuse.
leg a
C
32
Set up a proportion with
corresponding sides:
EF DF
=
BC AC
8 DF
=
16 32
8
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Example: Applying the Converse of
the Pythagorean Theorem
Converse of the Pythagorean Theorem
Is a triangle with sides of length 4, 7, and 8,
a right triangle?
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.
Solution
?
4 2 + 7 2 = 82
?
16 + 49 = 64
65 ≠ 64
No, it is not a right triangle.
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