5.5 – use inequalities in a triangle. mn p measure each side of the triangle in centimeters and...
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5.5 – Use Inequalities in a Triangle
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M N
P
Measure each side of the triangle in centimeters and each angle in degrees. Write these measurements on your triangle.
2.9cm 5.4cm
6.5cm56º 26º
98º
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Theorem:
If one side of the triangle is longer than the other side, then the angle opposite the longer side is larger than the angle opposite the shorter side
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Theorem:
If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
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1. List the sides and angles in order from smallest to largest.
Sides Angles
C
A
AB
BC
AC B
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Sides Angles
R
S
1. List the sides and angles in order from smallest to largest.
ST
RT
SR T
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55°
1. List the sides and angles in order from smallest to largest.
Sides Angles
H
J
KJ
KH
JH K
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33°
1. List the sides and angles in order from smallest to largest.
Sides Angles
E
D
DF
EF
DE F
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Construct a triangle with sides of 2, 2, and 6 cm.
2cm
6cm
Doesn’t work!
Why?
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6cm
2cm
Too short!
2cm
What if I tried 3, 3, and 6cm?
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6cm3cm 3cm
Still doesn’t work!
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Triangle Inequality Theorem:
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
a
b
ca + b > c
a + c > b
b + c > a
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2. Can a triangle be formed with sides 1, 2, and 5?
No, 1 + 2 > 5
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2. With 6, 7, and 11?
Yes, 6 + 7 > 117 + 11 > 66 + 11 > 7
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2. With 8, 12, and 20?
No, 8 + 12 > 20
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x + 9 > 21 x + 21 > 921 + 9 > xx > 12 30 > x
orx < 30
x > -12
12 < x < 30
3. What are the restrictions on x?
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Trick!
Subtract the two sides to find the lowest and add to find the highest.
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3. What are the possible values for the third side of a triangle with sides 9 and 12?
12 – 9 <
3 < x < 21
x < 12 + 9
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5. What are the possible values for the third side of a triangle with sides 14 and 20?
20 – 14 <
6 < x < 34
x < 20 + 14
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6. Solve the inequality AB + AC > BC for x.
3x + 1 + 4x – 2 > 5x + 7
7x – 1 > 5x + 7
2x – 1 > 7
2x > 8
x > 4