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Chapter 1
Section 9-4
The Geometry of Triangles:
Congruence, Similarity, and the
Pythagorean Theorem
9-4-1
© 2008 Pearson Addison-Wesley. All rights reserved
The Geometry of Triangles: Congruence,
Similarity, and the Pythagorean Theorem
• Congruent Triangles
• Similar Triangles
• The Pythagorean Theorem
9-4-2
© 2008 Pearson Addison-Wesley. All rights reserved
Congruent Triangles
B
E
A
D
F
C
9-4-3
© 2008 Pearson Addison-Wesley. All rights reserved
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.
9-4-4
© 2008 Pearson Addison-Wesley. All rights reserved
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.
9-4-5
© 2008 Pearson Addison-Wesley. All rights reserved
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.
9-4-6
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Proving Congruence (SAS)
Given:
Prove:
Proof
CE = ED
AE = EB
ACE  BDE
C
A
B
E
D
9-4-7
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Proving Congruence (ASA)
Given:
Prove:
Proof
ADB  CBD
ABD  CDB
ADB  CDB
B
A
C
D
9-4-8
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Proving Congruence (SSS)
Given:
Prove:
Proof
AD = CD
AB = CB
B
ABD  CDB
A
D
C
9-4-9
© 2008 Pearson Addison-Wesley. All rights reserved
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.
A
D
C
9-4-10
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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.
9-4-11
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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.
9-4-12
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Finding Side Length in
Similar Triangles
ABC is similar to DEF.
E
8
F
Find the length of side DF.
Solution
B
16
24
D
32
C
A
9-4-13
© 2008 Pearson Addison-Wesley. All rights reserved
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.
9-4-14
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Using the Pythagorean
Theorem
Find the length a in the right triangle below.
Solution
39
a
36
9-4-15
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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.
9-4-16
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Applying the Converse of
the Pythagorean Theorem
Is a triangle with sides of length 4, 7, and 8,
a right triangle?
Solution
.
9-4-17
© 2008 Pearson Addison-Wesley. All rights reserved