Download 5-5 Inequalities in Triangles

5-5 INEQUALITIES IN TRIANGLES (p. 273-279)
If a generous geometry teacher gives two different students packages of oatmeal from a
box containing exactly five packages, then he unfortunately can not distribute them
equally if he gives them all away. However, the amount that he gives to one student will
always be less than the original amount. This is an edible example of the Comparison
Property of Inequality.
Property Comparison Property of Inequality
If a = b + c and c > 0, then a > b.
This property allows you to prove a corollary to the Exterior Angle Theorem for
triangles. Recall that this theorem says that the measure of an exterior angle of a triangle
equals the sum of the measures of the two remote interior angles.
Example: Use the following diagram to answer this question. According to the Exterior
Angle theorem for triangles, m ACD  _______  _______ (fill in the blanks).
A
B
D
C
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 angles.
A
B
D
C
According to this corollary,
blanks)
m
ACD  ______ and
m
ACD  ______ (fill in the
Example: Discuss how you can use the exterior angle theorem for triangles and the
comparison property of inequality to prove this corollary.
Example: In the following diagram, AB
XY.
1. Why is 5  2 ?
2. Why is m 4 m 2 ?
3. Why is m 4 m 5 ?
B
5
Y
2
4
A
1
3
X
C
If two sides of a triangle are congruent, then the angles opposite those sides are
congruent. We learned this as Theorem 4-3 (Isosceles Triangle Theorem). What was the
short-cut way of remembering this theorem?
What do you think you can conclude if two sides of a triangle are not congruent?
Theorem 5-10
It two sides of a triangle are not congruent, then the larger angle lies opposite the
longer side.
B
A
If BC > AB, then
C
m
A  m C.
Example: In DOG, DO = 11, DG = 7, and OG = 13. List the angles from largest to
smallest. _______ > _______ > _______ (fill in the blanks)
If two angles of a triangle are congruent, then the sides opposite those angles are
congruent. We learned this as Theorem 4-4 (Converse of Isosceles Triangle Theorem).
What was the short-cut way of remembering this theorem?
What do you think you can conclude if two angles of a triangle are not congruent?
Theorem 5-11
If two angles of a triangle are not congruent, then the longer side is opposite the
larger angle.
X
Z
Y
If
m
Y  m X, then XZ  YZ.
There is no need to cover the indirect proof of Theorem 5-11.
Example: In CAT, m C  84, m A  32, and m T  64. List the sides from shortest
to longest.
_______ < _______ < _______ (fill in the blanks)
Not every set of three segments can form a triangle.
Example: Consider three segments with lengths of 6 cm, 8 cm, and 18 cm. Make a
sketch to show how these three segments can not “bridge the gap” to form a triangle.
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.
R
T
A
RA + AT > RT
RA + RT > AT
AT + RT > RA
Since the shortest distance between two points is a segment, the sum of the lengths of two
different noncollinear segments (that start and end at these two points) must be greater.
Example: Can a triangle have sides with the given lengths? Explain.
1. 3 cm, 3 cm, 6 cm
2. 8 ft, 19 ft, 12 ft
Do 4 on p. 276.
Example: In ABC, AB  7 and BC  18. Describe the possible lengths of AC.
Let x equal the length of AC and solve these inequalities.
1. x + 7 > 18
2. x + 18 > 7 (Note: You can really ignore this second inequality because the first
inequality actually includes this condition)
3. 7 + 18 > x (Note: This means the same as x  7  18)
Alternative method for the above example: Find the possible lengths of AC by using
simple subtraction and addition. To do this, find the difference of the two known lengths
and the sum of the two known lengths and insert these numbers in the following
inequality from left to right.
_______ < x < _______
Do 5 on p. 276.
Homework p. 276-279: 1,6,8,11,15,18,20,25,27,30a-c,31,32,36,42-44,51,55,59
31. The shortcut across the grass is shorter than the sum of the lengths of the other two
sides of the triangle (the two cement walkways).
42. Solve 6a  12  114
a = 17, so m P  46, m Q  68, and m QRP  66. PR is the longest length.