Triangle Inequality

Objective:

– Students make conjectures about the measures of opposite sides and angles of triangles.

If one side of a triangle is longer than the other sides, then its opposite angle is longer than the other two angles.

A

BC

Biggest Side is Opposite to Biggest AngleMedium Side is Opposite to Medium AngleSmallest Side is Opposite to Smallest Angle

49m<B is greater than m<C

6

Converse is true alsoBiggest Angle Opposite ______Medium Angle Opposite______Smallest Angle Opposite______

Angle B > Angle A > Angle C

So AC >BC > AB

If one angle of a triangle is longer than the other angles, then its opposite side is longer than the other two sides.

A

BC

49

6

84◦

47◦

Triangle Inequality

Objective:

– determine whether the given triples are possible lengths of the sides of a triangle

A

BC

49

6

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side

Inequalities in One Triangle

6

3 2

6

3 3

4 3

6

They have to be able to reach!!

Triangle Inequality Theorem

AB + BC > AC

AB + AC > BC

AC + BC > AB

A

BC

49

6

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 > 9Since the sum is greater than the third side, this is

a triangle

Triangle Inequality

Objective:

– Solve for a range of possible lengths of a side of a triangle given the length of the other two sides.

A triangle has side lengths of 6 and 12; what are the possible lengths of

the third side?

B

A

C

6 12

X = ?

1) 12 + 6 = 18

2) 12 – 6 = 6Therefore: 6 < X < 18

Examples

Describe the possible lengths of the third side of the triangle given the lengths of the other two sides.

5 in 12 in 18 ft. 12 ft.10 yd. 23 yd.