Triangle Inequalities

1
Triangle Inequalities

• Part 2

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Exterior Angle Inequality Theorem

• If an angle is an exterior angle of a triangle
  then its measure is greater than the measure of
  either of its corresponding remote interior
  angles.

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Example
4
Triangle Sides

• If one side of a triangle is longer than another
  side, then the angle opposite the longer side has
  a greater measure than the angle opposite the
  shorter side.

5
Examples

• List the Angles from largest to smallest

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Triangle Angles

• If one angle of a triangle has a greater measure
  than another angle, then the side opposite the
  greater angle is longer than the side opposite
  the lesser angle.

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Examples

• List the sides of triangle from largest to
  smallest

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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.
• Always check using the two smallest sides, they
  must be larger than the third. If this is true
  the numbers will represent a triangle.

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Example

• Do these numbers represent a triangle?
• 1.) 9, 7, 12
• Yes
• 2.) 5, 5, 10
• No
• 3.) 1, 4, 6
• No
• 4.) 6, 6, 2
• Yes

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Finding Range of Third Side

• If you are given two sides of a traingle you can
  determine the range that the third side must fall
  in.
• To find the smallest possible side length you
  subtract the larger side from the smaller side.
  The value you get can not be a side, however
  everything larger will work
• To find the largest your third side could be you
  add your two given sides, although this value
  will not work everything less than it will.

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Example

• If you have two sides of a triangle 4 in and 7 in
  what is the range for the possible third side, n.
  
• 3in lt n lt 11in
• If you have two sides of a triangle 8 in and 12
  in what is the range for the possible third side,
  n.
• 4in lt n lt 20in

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Hinge Theorem

• If two sides of one triangle are congruent two
  two sides of another triangle, and the included
  angles are not congruent, then the longest side
  is opposite the larger included angle.

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Example
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Converse of the Hinge Theorem

• If two sides of one triangle are congruent to two
  sides of another triangle, and the third sides
  are not congruent, then the larger included angle
  is opposite the longer third side.

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Example