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Geometry Concepts
Chapter 5 Triangle and Congruence
Identify the parts of a triangle
Classify triangles
Use the Angle Sum Theorem
Identify corresponding parts
Use
Use
Use
Use
SSS
SAS
ASA
AAS
Section 5.1 Classifying Triangles
Questions to think about:
•
Definition
Characteristics
TRIANGLE
Example
Nonexample
Definition
Characteristics
VERTEX
Example
Nonexample
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Classify Triangles
Classify by Angle
Classify by Side
ACUTE
All angles are acute
SCALENE
No sides congruent
OBTUSE
One obtuse angle
ISOCELES
At least two sides congruent
The congruent sides are
called legs and the third side
is the base
RIGHT
One right angle
EQUIANGULAR
All three angles congruent
EQUILATERAL
All sides are congruent
Examples…Classify each triangle by its angle and by it sides.
Triangle
Classification by Angle
Classification by Sides
(1.)
(2.)
(3.)
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(4.)
(5.)
(6.)
(7.)
(8.)
Examples…using algebra
(9.)
Find the measures of AB and
BC of isosceles triangle ABC if
∠A is the vertex angle.
(10.)
Find the measures of XY and
YZ of isosceles triangle XYZ if
∠X is the vertex angle.
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Section 5.2 Angles of a Triangle
Questions to think about:
THEOREM
ANGLE SUM THEOREM
5.1
The sum of the measure of the angles of a trianlge is 180.
Examples…
(11.)
Find m∠T in △RST.
(12.)
Find the value of each variable
in △DCE.
(13.)
Find m∠L in △MNL if m∠M=25 and m∠N=25.
(14.)
Find the value of each
variable in the figure.
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(15.)
Find m∠P in △MNP if m∠M=80 and m∠N=45.
(16.)
Find the value of each
variable △ABC.
THEOREM
5.2
The acute angles of a right triangle are
complementary.
Examples…
(17.)
Find m∠A and m∠B in right
triangle ABC.
(18.)
Find m∠J and m∠K in right
triangle JKL.
THEOREM
5.3
The measure of each angle of an equiangular triangle is 60.
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Section 5.4 Congruent Triangles
Questions to think about:
Definition
Characteristics
CONGRUENT
TRIANGLES
Example
Nonexample
Definition
Characteristics
CORRESPONDING
PARTS
Example
Nonexample
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Definition: Corresponding Parts of Corresponding Triangles
are Congruent
Characteristics
CONGRUENT
TRIANGLES (CPCTC)
Example
Nonexample
Examples…
(19.)
If △PQR ≅ △MLN, name the congruent angles and sides. Then draw the triangles, using arcs and slash
marks to show the congruent angles and sides.
(20.)
The corresponding
parts of two
congruent triangles
are marked on the
figure. Write a
congruence
statement for the
two triangles.
(21.)
The corresponding parts of
two congruent triangles
are marked on the figure.
Write a congruence
statement for the two
triangles.
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(22.)
△RST
is
congru
ent to
△XYZ.
Find
the
value of n.
(23.)
△UVW is congruent to △GHI. If m∠V = 90 and
m∠H = 3x + 15, find the value of x.
Section 5.5 and 5.6 SSS, SAS, ASA, AAS
Questions to think about:
POSTULATE
SSS- Side Side Side
5.1
If three sides of one triangle are congruent to three corresponding sides of another
triangle, then the triangles are congruent.
If AB ≅ DE , BC ≅ EF , and CA ≅ FD
then △ ABC ≅ △ DEF.
Examples…
(24.)
In two triangles, PQ ≅ ML , PR ≅ MN , and RQ ≅ NL . Write a congruence statement for the two triangles.
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Definition
Characteristics
INCLUDED ANGLE
Example
Nonexample
POSTULATE
SA S- Side Angle Side
5.2
If two sides and the included angle of one triangle are congruent to the corresponding
sides and included angle of
another trianle, then the
triangles are congruent.
If BO ≅ MA , ∠ O ≅ ∠ A and
OW ≅ AN
then △ BOW ≅ △ MAN.
Examples…
(25.)
Determine whether the triangles shown are congruent. If so, write a congruence
statement and explain why the triangles are congruent. If not, explain why not.
(26.)
Determine whether the triangles shown are congruent. If so, write a
congruence statement and explain why the triangles are congruent. If not,
explain why not.
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Definition
Characteristics
INCLUDED SIDE
Example
Nonexample
POSTULATE
5.3
A SA- Angle Side Angle
If two angles and the included
side of one triangle are
congruent to the corresponding
angles and included side of the
another triangle, then the
triangles are congurent.
Examples…
(27.)
In △PQR and △KJL, ∠ R ≅ ∠ K, RQ ≅ KL , and ∠ Q ≅ ∠ L. Write a congruence statement for the two triangles.
(28.)
In △DEF and △LMN, ∠ D ≅ ∠ N, DE ≅ NL , and ∠ E ≅ ∠ L. Write a congruence statement for the two triangles.
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POSTULATE
AA S- Angle Angle Side
5.4
If two angles and a
nonincluded side of one
triangle are congruent to the
corresponding two angles and
noninlcuded side of another
trianlge, then the triangles are
congruent.
Examples…
(29.)
△ABC and △DEF each have one pair of sides and one pair of
angles marked to show congruence. What other pair of angles
must be marked so that the two triangles are congruent AAS?
(30.)
△DEF and △LMN each have one pair of sides and one pair of angles marked to
show congruence. What other pair of angles must be marked so that the two
triangles are congruent AAS?
(31.)
What other pair of angles must be marked so that two triangles are congruent by ASA?
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Examples…Determine if the triangles are congruent by SSS, SAS, AAS, ASA. If not possible to prove congruent,
write not congruent.
(32.)
(33.)
(34.)
(35.)
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