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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 ...
Inequalities in One Triangle
• They have to be able to reach!!
6
3 2
6
3 3
4 3
6
Note that there is only one
situation tha...
Triangle Inequality Theorem
• AB + BC > AC
A
B
C
AB + AC > BC
AC + BC > AB
Triangle Inequality Theorem
A
B
C
Biggest Side Opposite Biggest Angle
Medium Side Opposite
Medium Angle
Smallest Side O...
Triangle Inequality Theorem
Converse is true also
Biggest Angle Opposite
_____________
Medium Angle Opposite
__________...
Example: List the measures of the sides of the
triangle, in order of least to greatest.
10x - 10 = 180
Solving for x:
B
A
...
Using the Exterior Angle Inequality
• Example: Solve the inequality if
AB + AC > BC
x + 3
x + 2
A
B
C
(x+3) + (x+ 2) > 3x ...
Example: Determine if the following lengths are
legs of triangles
A) 4, 9, 5
4 + 5 ? 9
9 > 9
We choose the smallest two of...
Example: a triangle has side lengths of 6 and
12; what are the possible lengths of the third
side?
B
A
C
6 12
X = ?
12 + 6...
Example: a triangle has side lengths of 6 and
12; what are the possible lengths of the third
side?
B
A
C
6 12
X = ?
12 + 6...
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5 2 triangle inequality theorem

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5 2 triangle inequality theorem

  1. 1. Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side
  2. 2. Inequalities in One Triangle • They have to be able to reach!! 6 3 2 6 3 3 4 3 6 Note that there is only one situation that you can have a triangle; when the sum of two sides of the triangle are greater than the third.
  3. 3. Triangle Inequality Theorem • AB + BC > AC A B C AB + AC > BC AC + BC > AB
  4. 4. Triangle Inequality Theorem A B C Biggest Side Opposite Biggest Angle Medium Side Opposite Medium Angle Smallest Side Opposite Smallest Angle 3 5 m<B is greater than m<C
  5. 5. Triangle Inequality Theorem Converse is true also Biggest Angle Opposite _____________ Medium Angle Opposite ______________ Smallest Angle Opposite _______________ B C A 65 30 Angle A > Angle B > Angle C So CB >AC > AB
  6. 6. Example: List the measures of the sides of the triangle, in order of least to greatest. 10x - 10 = 180 Solving for x: B A C Therefore, BC < AB < AC <A = 2x + 1 <B = 4x <C = 4x -11 2x +1 + 4x + 4x - 11 =180 10x = 190 X = 19 Plugging back into our Angles: <A = 39o ; <B = 76; <C = 65 Note: Picture is not to scale
  7. 7. Using the Exterior Angle Inequality • Example: Solve the inequality if AB + AC > BC x + 3 x + 2 A B C (x+3) + (x+ 2) > 3x - 2 3x - 22x + 5 > 3x - 2 x < 7
  8. 8. Example: Determine if the following lengths are legs of triangles A) 4, 9, 5 4 + 5 ? 9 9 > 9 We choose the smallest two of the three sides and add them together. Comparing the sum to the third side: B) 9, 5, 5 Since the sum is not greater than the third side, this is not a triangle 5 + 5 ? 9 10 > 9 Since the sum is greater than the third side, this is a triangle
  9. 9. Example: a triangle has side lengths of 6 and 12; what are the possible lengths of the third side? B A C 6 12 X = ? 12 + 6 = 18 12 – 6 = 6 Therefore: 6 < X < 18
  10. 10. Example: a triangle has side lengths of 6 and 12; what are the possible lengths of the third side? B A C 6 12 X = ? 12 + 6 = 18 12 – 6 = 6 Therefore: 6 < X < 18

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