Download 5_5_Inequalities_of_One_Triangle

5.5 Inequalities in One Triangle
Objectives:
 Students will analyze triangle measurements to decide
which side is longest & which angle is largest; students
will then apply the Triangle Inequality.
 Why? So you can find possible distances, as seen in
Ex. 39
 Mastery is 80% or better on 5-minute checks and
practice problems.
Skill Develop 1: Comparing Measurements of a
Triangle
largest angle
 You may discover a
relationship between the
positions of the longest
and shortest sides of a
triangle and the position
of its angles.
longest side
The diagrams illustrate Theorems 5.10
and 5.11.
shortest side
smallest angle
Skill Develop 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.
B
3
5
A
C
mA
> mC
Skill Develop Theorem 5.11
D
60°
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.
40°
E
F
EF
> DF
You can write the measurements
of a triangle in order from least to
greatest.
Skill Develop Ex. 1: Writing Measurements in
Order from Least to Greatest
Write the measurements
of the triangles from
least to greatest.
a. m G < mH < m J
JH < JG < GH
J
100°
45°
H
35°
G
Guided Practice Ex. 1: Writing Measurements in
Order from Least to Greatest
Write the measurements
of the triangles from
least to greatest.
QP < PR < QR
m R < mQ < m P
b.
R
8
Q
7
5
P
Quick Write---1 minute
 What have you discovered about the relationship
between and angles and side lengths in a Triangle?
 Explain.
Skill Develop Paragraph Proof – Theorem 5.10
A
Given►AC > AB
Prove ►mABC > mC
2
1
B
D
3
C
Use the Ruler Postulate to locate a point D on AC such that
DA = BA. Then draw the segment BD. In the isosceles
triangle ∆ABD, 1 ≅ 2. Because mABC =
m1+m3, it follows that mABC > m1. Substituting
m2 for m1 produces mABC > m2. Because m2
= m3 + mC, m2 > mC. Finally because mABC >
m2 and m2 > mC, you can conclude that mABC >
mC.
NOTE:
The proof of 5.10 in the slide previous uses the fact that
2 is an exterior angle for ∆BDC, so its measure is
the sum of the measures of the two nonadjacent
interior angles. Then m2 must be greater than the
measure of either nonadjacent interior angle. This
result is stated in Theorem 5.12
Theorem 5.12-Exterior Angle
Inequality-------Why?
 The measure of an exterior angle of a triangle is greater
than the measure of either of the two non adjacent
interior angles.
 m1 > mA and m1 > mB
A
1
C
B
Think…..Ink….Share
 DIRECTOR’S CHAIR. In the director’s chair shown,
AB ≅ AC and BC > AB. What can you conclude about
the angles in ∆ABC?

Because AB ≅ AC, ∆ABC is
isosceles, so B ≅ C.
Therefore, mB = mC.
Because BC>AB, mA >
mC by Theorem 5.10. By
substitution, mA > mB. In
addition, you can conclude
that mA >60°, mB< 60°,
and mC < 60°.
A
B
C
Objective 2: Using the Triangle
Inequality
 Not every group of three segments can be used to form
a triangle. The lengths of the segments must fit a
certain relationship.
Skill Develop Ex. 3: Constructing a Triangle
a.
b.
c.
2 cm, 2 cm, 5 cm
3 cm, 2 cm, 5 cm
4 cm, 2 cm, 5 cm
Solution: Try drawing triangles with the given side
lengths. Only group (c) is possible. The sum of the
first and second lengths must be greater than the
third length. Pay attention to the two smallest
measures then compare it to the 3rd.
Skill Develop Ex. 3: Constructing a Triangle
a.
b.
c.
2 cm, 2 cm, 5 cm
3 cm, 2 cm, 5 cm
4 cm, 2 cm, 5 cm
C
2
2
5
D
D
3
4
2
A
5
2
B
A
5
B
Skill Develop Theorem 5.13: Triangle Inequality
 The sum of the lengths
of any two sides of a
Triangle is greater than
the length of the third
side.
AB + BC > AC
AC + BC > AB
AB + AC > BC
A
C
B
Think….Ink…Pair Share
A
 Solve the inequality:
x+ 2
B
x+ 3
3x - 2
C
AB + AC > BC.
(x + 2) +(x + 3) > 3x – 2
2x + 5 > 3x – 2
5>x–2
7>x
Exit Slips
 A triangle has one side of
10 cm and another of 14
cm. Describe the possible
lengths of the third side
 SOLUTION: Let x
represent the length of the
third side. Using the
Triangle Inequality, you
can write and solve
inequalities.
x + 10 > 14
x>4
10 + 14 > x
24 > x
►So, the length of the third
side must be greater than
4 cm and less than 24 cm.
What was the Objective for today?
 Students will analyze triangle measurements to decide
which side is longest & which angle is largest; students
will then apply the Triangle Inequality.
 Why? So you can find possible distances, as seen in
Ex. 39
 Mastery is 80% or better on 5-minute checks and
practice problems.
Homework
 Page 331
 # 2 and 6-29