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Properties of Triangles
Unit 6
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5.5 Inequalities in One Triangle
State Standards for Geometry6. Know and use the
Triangle Inequality Theorem.
13. Prove relationships between angles in a polygon.
Lesson Goals• Use triangle measurements
to decide which side is longest or which angle is largest.
• Apply the Triangle Inequality Theorem to determine if 3 lengths can form a triangle, and find possible lengths of a 3rd side given two.
ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers
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A
B
Cthen is the shortest side.ABIf is the smallest angle, C
then is the longest side. ACIf is the largest angle, B
theorem
Triangle Angle-Side Relationships Theorem The longest side of a triangle is opposite the largest angle The shortest side of a triangle is opposite the
smallest angle.
largest angle
longest side
shorte
st sid
e
smallest angle
http://www.mathopenref.com/common/appletframe.html?applet=trianglebigsmall&wid=600&ht=300
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ACABBC
Write the measures for the sides of the triangle in order from least to greatest
A
B
C
111o
46o
23o
is opposite the smallest angle.BC
is opposite the middle-sized angle.AB
is opposite the largest angle.AC
example
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m U m T m V
Write the measures for the angles of the triangle in order from least to greatest
is opposite the shortest side.V
is opposite the middle-length side.T
is opposite the longest side.U
You Try
T
U
10
V
7
11
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theorem
Exterior Angle Inequality TheoremThe measure of an exterior angle of atriangle is greater than the measure of either of the two nonadjacent interior angles.
A
BC1
1m m A
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theorem
Exterior Angle Inequality TheoremThe measure of an exterior angle of atriangle is greater than the measure of either of the two nonadjacent interior angles.
A
BC1
1m m A
1m m C
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14
3
2
3 54 6
example
List all angles whose measures are less than 1.m
5
6
7
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Write an equation or inequality to describe the relationship between the measures of all angles.
ao
doco
bo
180a b c
a b d
180c d d a
d b
example
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Triangle Inequality TheoremThe sum of the lengths of any two sides of a triangle is greater than the length of the third side.
A
BC
theorem
A C
CA B
AB BC AC
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Triangle Inequality TheoremThe sum of the lengths of any two sides of a triangle is greater than the length of the third side.
A
BC
theorem
AB AC BC
B C
CB A
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Triangle Inequality TheoremThe sum of the lengths of any two sides of a triangle is greater than the length of the third side.
A
BC
theorem
A B
CA B
AC BC AB
http://www.mathopenref.com/common/appletframe.html?applet=triangleinequality2&wid=600&ht=200
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Can a triangle be constructed with sides of the following measures?
5, 7, 8
By the Triangle Inequality Theorem, the sum of the measures
of any two sides must by greater than the third side.
5 + 7 > 8 5 + 8 > 7 7 + 8 > 5
example
Yes
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Can a triangle be constructed with sides of the following measures?
3, 6, 10
3 + 6 > 10
example
No
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Can a triangle be constructed with sides ofthe following measures?
By the Triangle Inequality Theorem, the sum of the measures
of any two sides must by greater than the third side.
9 + 5 > 11
A triangle can be constructed.
You Try
9 in, 5 in , 11 in
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Solve the inequality.
exampleAB AC BC
5 2 1 10x x x
3 4 10x x
4 4 10x
4 6x 3
2x
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A triangle has one side of 8 cm and another of 17 cm. Describe the possible lengths of the third side.
By the Triangle Inequality Theorem, the sum of the measures
of any two sides must by greater than the third side.
x + 8 > 17 x + 17 > 8 8 + 17 > x
817
x
x > 9 x > -9 25 > x9 < x x < 25
example
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x < 27
A triangle has one side of 11 in and another of 16 in. Describe the possible lengths of the third side.
By the Triangle Inequality Theorem, the sum of the measures
of any two sides must by greater than the third side.
x + 11 > 16 x + 16 > 11 11 + 16 > x
1116
x
x > 5 x > -5 27 > x5 < x
example
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Today’s Assignment
p. 298: 1 – 6, 9, 12, 13, 16, 17, 20, 23 – 25, 27, 28
(worksheet in packet)