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Chapter 4
Congruent Triangles
Classifying Triangles
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Triangle – a three sided polygon
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Sides are segments
Vertices are points
Has three angles
Two ways to classify:
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Angles
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All triangles have at least two acute angles, but the third
angle is the classifying angle
Number of congruent sides they have
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Equal number of hash marks are drawn on corresponding
sides to indicate that sides are congruent
Classifying Triangles by Angles
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Acute Triangle
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A triangle where all of the angles are acute
All angle measures are less than 90 degrees
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An acute angle with all angles congruent is an
equiangular triangle
Classifying Triangles by Angles
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Obtuse Triangle
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A triangle where one angle that is obtuse
One angle measure is greater than 90 degrees
Classifying Triangles by Angles
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Right Triangle
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A triangle where one angle is right
One angle measure is equal to 90 degrees
Classifying Triangles by Sides
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Scalene Triangle
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A triangle where no two sides are congruent
Classifying Triangles by Sides
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Isosceles Triangle
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A triangle with at least two sides are congruent
Classifying Triangles by Sides
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Equilateral Triangle
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A triangle where all the sides are congruent
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An equilateral triangle is a special kind of isosceles
triangle
Example One
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Identify the indicated type of triangles in the
figure:
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Isosceles Triangles
Scalene Triangles
Acute Triangles
Right Triangles
You Do It
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Identify the indicated triangles in the figure if
UV = VX = UX.
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Isosceles Triangles
Scalene Triangles
Obtuse Triangles
Acute Triangles
Example Two
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Find x and the measure of each side of
equilateral triangle RST if RS = x + 9,
ST = 2x, and RT = 3x – 9
You Do It
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Find d and the measure of each side of
equilateral triangle KLM if KL = d + 2,
LM = 12 – d, and KM = 4d – 13.
Let’s Try another one 
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Find x and the measure of each side of
equilateral triangle HKT if HK = x + 7,
HT = 4x – 8, and KT = 2x + 2
Using Distance Formula?
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Find the measures of the sides of ΔDEC.
Classify the triangle by sides.
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D(3, 9)
E(-5, 3)
C(2, 2)
You Do It
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Find the measures of the sides of ΔRST.
Classify the triangle by sides.
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R(-1, 3)
S(4, 4)
T(8, -1)
Angle Sum Theorem
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The sum of the measures of a triangle is 180.
– Example:
 Angle
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W + Angle X + Angle Y = 180
If we know the measures of two angles of a
triangle, then we can find the third.
Example
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Find the missing angle measures.
You Do It
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Find the missing angle measures.
Third Angle Theorem
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If two angles of one triangle are congruent to
two angles of a second triangle, then the
third angles of the triangles are congruent.
Exterior Angles
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Each angle of a triangle has an exterior
angle.
Exterior Angle: formed by one side of a
triangle and the extension of another side
Remote Interior Angles: the interior angles of
the triangle not adjacent to a given exterior
angle
Exterior Angle Theorem
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The measure of an exterior angle of a
triangle is equal to the sum of the measures
of the two remote interior angles.
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Example:
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Angle YZP = Angle X + Angle Y
Example
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Find the measure of each numbered angle in
the figure
You Do It
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Find the measure of each numbered angle in
the figure
Corollary
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Statements that can be used to prove
theorems
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Corollary 1:
 The acute angle of a right triangle are
complementary
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Corollary 2:
 There can be at most one right or obtuse angle
in a triangle
Classwork/Homework
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Pages 180 – 181
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14 – 18 (even)
22 – 28 (even)
32 – 34 (even)
Pages 189 – 190
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12 – 26 (even)
32 and 34
Congruent Triangles
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Triangles that are the same size and shape
All three angles and three sides must be
congruent
Corresponding parts of the triangles must be
congruent
Definition: Two triangles are congruent if and
only if their corresponding parts are
congruent. Also known as CPCTC.
Properties
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Just like segments and angles, congruence
of triangles is reflexive, symmetric, and
transitive.
If you slide, flip, or turn a triangle, the size
and shape do not change.
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This is called the congruence transformations
Side-Side-Side Congruence
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If the sides of one triangle are congruent to
the sides of a second triangle, then the
triangles are congruent.
Abbreviation: SSS
Example
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Determine whether ΔRTZ is congruent ΔJKL
for R(2, 5), Z(1,1), T(5, 2), L(-3, 0), K(-7, 1),
and J(-4, 4). Explain.
You Do It
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Determine whether ΔWDV is congruent to
ΔMLP if W(-7, -4), D(-5, -1), V(-1, -2), M(4, 7), L(1, -5), and P(2, -1). Explain.
Side-Angle-Side Congruence
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If two sides and the included angle of one
triangle are congruent to two sides and the
included angle of another triangle, then the
triangles are congruent.
Abbreviation: SAS
Example
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Determine which postulate can be used to
prove that the triangles are congruent. If it is
not possible to prove that they are congruent,
write not possible.
Angle-Side-Angle Congruence
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If two angles and the included side of one
triangle are congruent to two angles and the
included side of another triangle, then the
triangles are congruent
Abbreviation: ASA
Angle-Angle-Side Congruence
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If two angles and a nonincluded side of one
triangle are congruent to the corresponding
two angles and side of a second triangle,
then the two triangles are congruent.
Abbreviation: AAS
Methods to Prove Triangle Congruence
Definition of Congruent Triangles
All corresponding parts of one triangle are
congruent to the corresponding parts of the
other triangle.
SSS
The three sides of one triangle must be
congruent to the three sides of the other
triangle
SAS
Two sides and the included angle of one
triangle must be congruent to two sides and
the included angle of the other triangle.
ASA
Two angles and the included side of one
triangle must be congruent to two angles and
the included side of the other triangle
AAS
Two angles and a nonincluded side of one
triangle must be congruent to two angles and
side of the other triangle
Example
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Suppose the redwood supports of a chapel,
TU and TV, measure 3 feet, TY = 1.6 feet,
and the measurement of angle U and the
measurement of angle V are 31. Determine
whether ΔTYU is congruent to ΔTYV. Justify
your answer.
You Do It
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When Ms. Gomez puts her hands on her
hips, she forms two triangles with her upper
body and arms. Suppose her are lengths AB
and DE measure 9 inches, and AC and EF
measure 11 inches. Also suppose that you
are given BC = DF. Determine whether
ΔABC is congruent to ΔEDF.
Classwork/Homework
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Worksheet
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Please complete both sides for tomorrow
Remember you have a quiz tomorrow on sections
4.1 – 4.5
Quiz Time
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You have 20 – 25 minutes to take your quiz.
If you have any questions, please ask
Good luck and take your time 
Congruence in Right Triangles
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Just like regular triangles, right triangles have
congruencies.
They are:
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Leg-Leg Congruence
Hypotenuse-Angle Congruent
Leg-Angle Congruence
Hypotenuse-Leg Congruence
Leg-Leg Congruence
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If the legs of one right triangle are congruent
to the corresponding legs of another right
triangle, then the triangles are congruent.
Abbreviation: LL
Hypotenuse-Angle Congruence
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If the hypotenuse and acute angle of one
right triangle are congruent to the
hypotenuse and corresponding acute angle
of another right triangle, then the triangles
are congruent.
Abbreviation: HA
Leg-Angle Congruence
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If one leg and an acute angle of one right
triangle are congruent to the corresponding
leg and acute angle of another right triangle,
then the triangles are congruent.
Abbreviation: LA
Hypotenuse-Leg Congruence
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If the hypotenuse and a leg of one right
triangle are congruent to the hypotenuse and
corresponding leg of another right triangle,
then the triangles are congruent.
Abbreviation: HL
Isosceles Triangles Theorem
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If two sides of a triangle are congruent, then
the angles opposite those sides are
congruent
Example
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If GH = HK, HJ = JK, and the measurement
of angle GJK is 100, what is the measure of
angle HGK?
You Do It
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If DE = CD, BC = AC, and the measurement
of angle CDE is 120, what is the measure of
angle BAC?
What is the converse of the Isosceles
Triangle Theorem?
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Isosceles Triangle Theorem: If two sides of a
triangle are congruent, then the angles
opposite those sides are congruent.
Converse: If two angles of a triangle are
congruent, then the sides opposite those
angles are congruent.
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Abbreviation: Conv. Of Isos. Δ Th.
Example
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Name two congruent angles
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Name two congruent segments
You Do It
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Name two congruent angles
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Name two congruent angles
Properties of Equilateral Triangles
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A triangle is equilateral if and only if it is
equiangular.
Each angle of an equilateral triangle
measures 60º
Example
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ΔEFG is equilateral, and EH bisects angle E.
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Find the measurement of angle 1 and angle 2
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Find x
You Do It
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Using the same figure as above, draw EJ so
that EJ bisects angle 2 and J lies on FG.
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Find the measurement of HEJ and the
measurement of EJH
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Find the measurement of EJG
Coordinate Proof
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Uses figures in the coordinate plane and
algebra to prove geometric concepts.
Steps:
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Use the origin as a vertex or center of the figure
Place at least on side of a polygon on an axis
Keep the figure within the first quadrant if possible
Use coordinates that make computations as
simple as possible
Example
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Position and label isosceles triangle JKL on a
coordinate plane so that base JK is a units
long
You Do It