Congruent%20Triangles

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Congruent Triangles

• 4-1 Classifying Triangles
• 4-2 Angle Measure in Triangles
• 4-3 Congruent Triangles
• 4-4 4-5 Tests for Congruent Triangles
• 4-7 Isosceles Triangles

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2
4-1 Classifying Triangles
The Parts of a Triangle
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4-1 Classifying Triangles
Classifying by Angles
Acute Triangle
Obtuse Triangle
In an Acute Triangle all the angles are acute,
less than 90.
In an Obtuse Triangle there is only one obtuse
angle, an angle greater than 90.
Right Triangle
A Right Triangle has one 90 degree angle.
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4
4-1 Classifying Triangles
Right Triangles
Leg
Hypotenuse
Equiangular Triangles
Leg
In an Equiangular Triangle all three angles are
congruent to one another.
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4-1 Classifying Triangles
Classifying by Sides
Isosceles Triangle
Scalene Triangle
In an Isosceles Triangle at least two of the
sides are congruent.
In an Scalene Triangle all the sides have
different lengths
Equilateral Triangle
In an Equilateral Triangle all the sides are
congruent to each other
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4-1 Classifying Triangles
Isosceles Triangles
In an Isosceles Triangle the two congruent sides
are called legs and form the vertex angle. The
side opposite the vertex angle is called the
base. The base and each leg form two base angles.
The base angles of an isosceles triangle are
congruent.
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Section 4-2
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4-2 Angle Measures in Triangles
Theorem
The sum of the measures of the angles of a
triangle is 180.
Given ?EFG m?E m?F m?G 180.
Topic 4
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8
4-2 Angle Measures in Triangles
Theorem
If two angles of one triangle are congruent to
two angles of another triangle, then the third
angles of the triangles are congruent.
Given ?EFG and ?PQR If ?E ? ?R, and ?F ? ?Q,
then ?G ? ?P
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4-2 Angle Measures in Triangles
Definitions
An Exterior Angle is formed by one side of a
triangle and another side extended. The angles
of the triangle not adjacent to a given exterior
angle are called Remote Interior Angles.
Given ?ZXY an interior angle of ?XYZ. ?WXZ is
the corresponding exterior angle. ?Z and ?Y are
the resulting remote interior angles
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10
4-2 Angle Measures in Triangles
Theorem
The measure of an exterior angle of a triangle is
equal to the sum of the measures of its two
remote interior angles.
m?WXZ m?Z m?Y
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4-2 Angle Measures in Triangles
Corollary
Given Right Triangle ?ABC
The acute angles of a right triangle are
complementary.
Corollary
There can be at most one right or obtuse angle in
a triangle.
m?A m?B 90.
A Corollary is a statement that follows
directly from another theorem and that can be
easily proved from that theorem.
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4-3 Congruent Triangles
Definition
Two triangles are congruent if and only if their
corresponding parts are congruent.
?ABC ? ?XYZ if and only if the three
corresponding sides are congruent and the three
corresponding angles are congruent
Angles ?A ? ?X ?B ? ?Y ?C ? ?Z
This definition can be abbreviated CPCTC which
means Corresponding Parts of Congruent Triangles
are Congruent.
Topic 4
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4-3 Congruent Triangles
Theorem
Congruence of Triangles is Reflexive, Symmetric
and Transitive.
Reflexive Property Symmetric Property Transitive Property
?ABC ? ?ABC If ?ABC ? ?DEF, then ?DEF ? ?ABC If ?ABC ? ?DEF, and ?DEF ? ?GHI, then ?ABC ? ?GHI
Section 4-4
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4-4 4-5 Tests for Congruent Triangles
Side-Side-Side (SSS) Postulate
If the three sides of one triangle are congruent
to the three sides of another triangle, then the
triangles are congruent.
Topic 4
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4-4 4-5 Tests for Congruent Triangles
Side-Angle-Side (SAS) Postulate
If two sides and an included angle of one
triangle are congruent to two sides and an
included angle of another triangle, then the
triangles are congruent.
?
?
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4-4 4-5 Tests for Congruent Triangles
Angle-Side-Angle (ASA) Postulate
If two angles and an included side of one
triangle are congruent to two angles and an
included side of another triangle, then the
triangles are congruent.
?
?
?
?
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4-4 4-5 Tests for Congruent Triangles
Angle-Angle-Side (AAS) Postulate
If two angles and an non-included side of one
triangle are congruent to the corresponding two
angles and side of another triangle, then the
triangles are congruent.
?
?
?
?
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18
4-4 4-5 Tests for Congruent Triangles
Sample Proof
1. ?R and ?T are Right Angles. ?1 ? ?2
1. Given
2. ?R ? ?T
2. Right Angles are Congruent
3. SV ? SV
3. Reflexive
4. ?RSV ? ?TSV
4. AAS
5. CPCTC
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Section 4-7
19
4-7 Isosceles Triangles
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then
the angles opposite those sides are congruent.
Theorem
If two angles of a triangle are congruent, then
the sides opposite those angles are congruent.
Topic 4
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4-7 Isosceles Triangles
Corollary
A triangle is equilateral if and only if it is
equiangular
Corollary
Each angle of an equilateral triangle measures 60
degrees.
If ?XYZ is equilateral then, m?X m?Y m?Z 60.
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Applying Congruent Triangles