Transcript Triangle Inequality Theorem

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.
Biggest Side is Opposite to Biggest Angle
Medium Side is Opposite to Medium Angle
Smallest Side is Opposite to Smallest Angle
A
m<B is greater than m<C
C
9
4
6
B
If one angle of a triangle is longer than the
other angles, then its opposite side is longer
than the other two sides.
Converse is true also
Biggest Angle Opposite ______
Medium Angle Opposite______
Smallest Angle Opposite______
A
Angle B > Angle A > Angle C
So AC >BC > AB
9
4
84◦
47◦
C
6
B
Triangle Inequality
Objective:
–
determine whether the given triples are possible
lengths of the sides of a triangle
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
9
C
4
6
B
Inequalities in One Triangle
They have to be able to reach!!
3
2
4
3
6
3
3
6
6
Triangle Inequality Theorem
AB + AC > BC
AB + BC > AC
A
AC + BC > AB
9
C
4
6
B
Example: Determine if the following
lengths are legs of triangles
A)
4, 9, 5
B)
9, 5, 5
We choose the smallest two of the three sides and add
them together. Comparing the sum to the third side:
4+5 ? 9
5+5 ? 9
9>9
10 > 9
Since the sum is
not greater than
the third side,
this is not a
triangle
Since 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
6
12
A
X=?
1) 12 + 6 = 18
2) 12 – 6 = 6
Therefore:
C
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
10 yd.
23 yd.
18 ft. 12 ft.