Triangle Inequalities

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• Triangle Inequalities

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Triangle Inequality

• The smallest side is across from the smallest
  angle.
• The largest angle is across from the largest side.

BC 3.2 cm
AB 4.3 cm
AC 5.3 cm
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Triangle Inequality examples
For the triangle, list the angles in order from
least to greatest measure.
4 cm
6 cm
5 cm
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If one side of a triangle is longer than the
other side, then the angle opposite the longest
side is _______ than the angle opposite the
shorter side.
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If one angle of a triangle is larger than another
angle, then the side opposite the larger angle is
_______ than the side opposite the smaller angle.
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Write measurements in order from least to greatest
Example 1
Write measurements of the triangle in order from
least to greatest.
a.
b.
Solution
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Write measurements in order from least to greatest
Example 1
Write measurements of the triangle in order from
least to greatest.
a.
b.
Solution
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Checkpoint. Write the measurements of the
triangle in order from least to greatest.

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Checkpoint. Write the measurements of the
triangle in order from least to greatest.

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Triangle Inequality examples
For the triangle, list the sides in order from
shortest to longest measure.
(7x 8) (7x 6 ) (8x 10 )
180 22 x 4 180 22x 176 X 8
mltC 7x 8 64 mltA 7x 6 62 mltB 8x
10 54
54
64
62
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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.
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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 gt c a c gt b b c gt a
Example
Determine if it is possible to draw a triangle
with side measures 12, 11, and 17.
12 11 gt 17 ? Yes 11 17 gt 12 ? Yes 12 17 gt
11 ? Yes
Therefore a triangle can be drawn.
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Finding the range of the third side

• Since the third side cannot be larger than the
  other two added together, we find the maximum
  value by adding the two sides.

Since the third side and the smallest side cannot
be larger than the other side, we find the
minimum value by subtracting the two sides.
Example
Given a triangle with sides of length 3 and 8,
find the range of possible values for the third
side.
The maximum value (if x is the largest side of
the triangle) 3 8 gt x 11 gt x
The minimum value (if x is not that largest side
of the ?) 8 3 gt x 5gt x
Range of the third side is 5 lt x lt 11.
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Find possible side lengths
Example 2
A triangle has one side of length 14 and another
of length 10. Describe the possible lengths of
the third side.
Solution
Let x represent the length of the third side.
Draw diagrams to help visualize the small and
large values of x. Then use the Triangle
Inequality Theorem to write and solve
inequalities.
Small values of x
Large values of x
The length of the third side must be
_______________________________.
greater than 4
and less than 24
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Checkpoint. Complete the following exercise

1. A triangle has one side 23 meters and another of
   17 meters. Describe the possible lengths of the
   third side.

Small values of x
Large values of x
The length of the third side must be
greater than 6 meters
or less than 40 meters
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Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is
equal to the sum of the two non adjacent interior
angles.
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Relate exterior and interior angles
Example 3
Solution
So, by the Exterior Angle Inequality Theorem,
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Checkpoint. Complete the following exercise