Download Inequalities in One Triangle

Section5.6
Inequalities in One Triangle
Objectives:
To use inequalities involving angles and sides of triangles
To use inequalities involving angles and sides of triangles
MA. 912.G.4.7
Corollary to the
Triangle Exterior Angle
Theorem
The measure of an exterior angle of a triangle is
greater than the measure of each of its remote interior
angle.
IF:  1 is an exterior angle
3
2
THEN: m1 > m2 and
m1 > m3
1
We learned in chapter 3 that angles 2 and 3 must add up to
be the measure of angle 1.
Theorem 5-10
If one side of a triangle is longer than
another side, then the angle opposite the
longer side is larger than the angle
opposite the shorter side.
m∠A > m∠C
Theorem 5-11
If one angle of a triangle is larger than another
angle, then the side opposite the larger angle is
longer than the side opposite the smaller angle.
FE > DF
Example # 1
Inequalities Within a Triangle
The longest side is
BC
So, the largest angle is
The largest angle is
A
L
So, the longest side isMN
Example # 2: How to write the
measurements.
• What these two theorems mean, is that you can
put the measurements of a triangle in order,
either from least to greatest, or greatest to least.
Example:
Write the measures of this
triangle in order, from least
to greatest.
Angles: Angle E, Angle D, Angle F
Sides: DF, FE, DE
80˚
Theorem 5-12 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
AB + BC > AC
AC + BC > AB
AB + AC > BC
B
C
Steps to determine side length
1. To determine if three side lengths will form a
triangle, add the lengths of the shortest two
sides.
2. If the sum is more than the length of the third side
then the three lengths will form a triangle.
3. If the sum is less than or equal to the length of
the third side, then the three lengths will not form
a triangle.
Example #3
1. Can a triangle have
sides with the
given lengths?
Explain
2. Can a triangle have
sides with the given
lengths? Explain
8m, 10m, 19m
2 yds, 9 yds, 10 yds
No, because 8+10=18
which is not
greater than 19.
Yes, because 2+9=11
which is greater
than 10.
You Try:
Can a triangle have sides with the given
lengths? Explain
A. 5, 6, 8
B. 5, 3, 8
C. 6, 3, 5
D. 3, 6, 8
E. 5, 6, 11
F. 5, 11, 8
A. Yes
B. No
C. Yes
D. Yes
E. No
F. Yes
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 4: Given a triangle with sides of
lengths 3 and 8, find the range of
possible values for the third side.
The maximum value is:
8+3>x
11>x
The minimum value is:
8-3<x
5<x
Range of the third side is:
5<x<11
You Try:
1. Given a triangle with
sides of length 5 and
16, find the range of
possible values for
the third side.
11 < x < 21
2. Given a triangle with
sides of length 6 and
6, find the range of
possible values for
the third side
0 < x < 12
That’s a wrap…..
• Is there anything that you are still have trouble
with in Chapter 5?
• Is there anything that you need more help with?