Topic 5-7
Inequalities in one triangle
How many different triangles can we make using these six pieces?
1. What are your
guesses?
2. What guess is too
low? Too high?
3. What strategies are
you using as you
think about how
many triangles you
can make?
4. What is the
relationship between
the side lengths?
2
Triangle Inequality Theorem:
The sum of the lengths of any two sides of a triangle is
greater than the length of the third side. B
a+b>c
a
c
a+c>b
b+c>a
A
C
b
Example: Determine if it is possible to draw a triangle with side
measures 12, 11, and 17. 12 + 11 > 17  Yes
Therefore a triangle can be drawn.
11 + 17 > 12  Yes
12 + 17 > 11  Yes
Example
Determine if it is possible to draw a triangle with side
measures 13, 6, and 7.
13 + 6 > 7 Yes
13 + 7 > 6 Yes
7 + 6 = 13 No
Therefore a triangle cannot be drawn.
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 The minimum value (if x is not that largest
side of the triangle)
3+8>x
side of the ∆)
8–3>x
11 > x
Range of the third side is 5 < x < 11.
5> x
Example
The lengths of two sides of a triangle are 6 and 8. Write
the inequality that represents the possible lengths for the
third side.
The maximum value (if x is the largest
side of the triangle)
6+8>x
The minimum value (if x is not that largest
side of the ∆)
8–6>x
14 > x
Range of the third side is 2 < x < 14
2>x
Triangle Inequality

The smallest side is across from the smallest angle.
A is thesmallest angle,  BC is thesmallest side.

The largest angle is across from the largest side.
B is the largest angle,  AC is the largest side.
B
89
54
37
A
C
Triangle Inequality – examples…
For the triangle, list the angles in order from least to greatest measure.
B
A
AB is the smallest side  C smallest angle.
5 cm
BC is thel arg est side  Ais the l arg est angle.
Angles in order from least to greatest  C , B, A
C