Chapter 4: Congruent Triangles

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Chapter 4 Congruent Triangles

• Lesson 1 Classifying Triangles

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Classifying Triangle by Angles

• Acute Triangle all of the angles are acute
• Obtuse Triangle one angle is obtuse, the other
  two are acute
• Right Triangle one angle is right, the other two
  are acute
• Equiangular Triangle all the angles are 60
  degrees

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Classifying Triangles by Sides

• Scalene Triangle all sides are different
  measures
• Isosceles Triangle at least two sides have the
  same measure
• Equilateral Triangle all sides have the same
  measure

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3
5
vertex angle formed by the two congruent sides
of an isosceles triangle base the side of an
isosceles triangle not congruent to the others
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ALGEBRA Find x and the measure of each side of
equilateral triangle ABC if AB 6x 8, BC 7
x, and AC 13 x.
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• Find the measure of each side of Triangle JKL and
  classify the triangle based on its sides.
• J(-3, 2) K(2, 1) L(-2, -3)

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Chapter 4 Congruent Triangles

• Lesson 2 Angles of Triangles

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• The sum of the measures of the angles of a
  triangle is always 180 degrees.
• The acute angles of a right triangle are
  complementary
• There can be at most one right or one obtuse
  angle in a triangle

• Third Angle Theorem
• If two angles of one triangle are congruent to
  two angles of another triangle, then the third
  angles of the triangles are also congruent.

X
A
Y
B
Z
C
If A X, and B Y, then
C Z.
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Interior and Exterior Angles of Triangles

• Exterior angle formed by one side of a triangle
  and the extension of another side
• The interior angles farthest from the exterior
  angle are its remote interior angles. (remote
  interior angles are not adjacent to the exterior
  angle)

Exterior angle
Remote interior angles
An exterior angle is equal to the sum of its
remote interior angles. ex 1 2 4
2
1
3
4
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Anticipation Guide read each statement. State
whether the sentence is true or false. If the
statement is false- rewrite it with the correct
term in place of the underlined word

• The acute angles of a right triangle are
  supplementary
• The sum of the measures of the angles of any
  triangle is 100
• A triangle can have at most one right angle or
  acute angle
• If two angles of one triangle are congruent to
  two angles of another triangle, then the third
  angle of the triangles are congruent
• The measure of an exterior angle of a triangle is
  equal to the difference of the measures of the
  two remote interior angles
• If the measures of two angles of a triangle are
  62 and 93, then the measure of the third angle is
  35
• An exterior angle of a triangle forms a linear
  pair with an interior angle of the triangle

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SOFTBALL The diagram shows the path of the
softball in a drill developed by four players.
Find the measure of each numbered angle.
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Find the measure of each numbered angle.
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GARDENING Find the measure of ?FLW in the fenced
flower garden shown.
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The piece of quilt fabric is in the shape of a
right triangle. Find the measure of ?ACD.
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Find the measure of each numbered angle.
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Find m?3.
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Chapter 4 Congruent Triangles

• Lesson 6 Isosceles Triangles

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Isosceles Triangles

• .

Vertex Angle
- If two sides of a triangle are congruent, the
two angles opposite of them are also congruent
leg
leg
-If two angles of a triangle are congruent, then
two sides opposite of them are also congruent
Base angles
- If a triangle is equilateral, it is also
equiangular
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• A. Find m?R.

B. Find PR
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• A. Find m?T.

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ALGEBRA Find the value of each variable
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Chapter 4 Congruent Triangles

• Lesson 3 Congruent Triangles

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Definition of Congruent Triangles

• Congruent triangles are triangles with exactly
  the same size and shape
• CPCTC Corresponding Parts of Congruent Triangles
  are Congruent
• Two triangles are congruent if and only if their
  corresponding parts are congruent

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Corresponding Parts
A

• Corresponding parts have the same congruence
  markings
• AB HI
• AC HJ
• BC IJ
• A H
• B I
• C J

B
C
H
I
J
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Congruence Transformations

• Slide or Translation the triangle is in the same
  position farther down, up, or across the page
• Turn or Rotation the triangle is spun around a
  point (usually one of the angles)
• Flip or reflection the triangle is shown in a
  mirror image across a line of symmetry

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Write a congruence statement for the triangles.
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Name the corresponding congruent angles for the
congruent triangles.
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In the diagram, ?ITP ? ?NGO. Find the values of
x and y.
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In the diagram, ?FHJ ? ?HFG. Find the values of
x and y.
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Find the missing information in the following
proof.
Prove ?QNP ? ?OPN
Proof
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Write a two-column proof.
Prove ?LMN ? ?PON
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Chapter 4 Congruent Triangles

• Lesson 4 and 5 Proving Congruence- SSS, SAS,
  ASA, AAS, and HL

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SSS

• Side-Side-Side
• If all three sets of corresponding sides are
  congruent, the triangles are congruent

A
M
B
C
O
N
ABC MNO
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SAS

• Side-Angle-Side
• If two corresponding sides and the included
  angles of two triangles are congruent, then the
  triangles are congruent

The included angle is the angle between the
congruent sides
X
F
Y
Z
G
H
XYZ FGH
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ASA

• Angle-Side-Angle
• If two sets of corresponding angles and the
  included sides are congruent, then the triangles
  are congruent

The included side is the side between the two
congruent angles
J
R
L
K
T
S
JKL RST
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AAS

• Angle-Angle-Side
• If two sets of corresponding angles and one of
  the corresponding non-included sides are
  congruent, then the triangles are congruent

T
E
G
F
V
U
EFG
TUV
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HL

• Hypotenuse-Leg
• If the hypotenuse and one set of corresponding
  legs of two right triangles are congruent, then
  the triangles are congruent

C
R
D
H
A
M
CDH
RAM
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Determine if the triangles are congruent. If they
are, write the congruence statement.
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• Determine whether ?ABC ? ?DEF for A(5, 5), B(0,
  3), C(4, 1), D(6, 3), E(1, 1), and F(5, 1).

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Determine if the triangles are congruent. If they
are, write the congruence statement.
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Determine which postulate can be used to prove
that the triangles are congruent. If it is not
possible to prove congruence, choose not possible.
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Write a two column proof.
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