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LESSON
YO
LESSON
4.3
4.3
Triangle Inequalities
How long must each side of this drawbridge be so that the bridge spans the river
when both sides come down?
PLANNING
Readers are plentiful, thinkers
are rare.
LESSON OUTLINE
HARRIET MARTINEAU
One day:
25 min Investigation
10 min Sharing
5 min Closing
5 min Exercises
MATERIALS
Drawbridges over the
Chicago River in
Chicago, Illinois
The sum of the lengths of the two parts of the drawbridge must be equal to or
greater than the distance across the waterway. Triangles have similar requirements.
You can build a triangle using one blue rod, one green rod, and one red rod. Could
you still build a triangle if you used a yellow rod instead of the green rod? Why or
why not? Could you form a triangle with two yellow rods and one green rod? What
if you used two green rods and one yellow rod?
construction tools
protractors
sticks or uncooked spaghetti, optional
Triangle Exterior Angle
Conjecture (W), optional
Sketchpad activity Triangle
Inequalities, optional
TEACHING
This lesson concerns three properties of triangles: the triangle inequality, the side-angle inequality,
and the exterior angle property.
How can you determine which sets of three rods can be arranged into triangles and
which can’t? How do the measures of the angles in the triangle relate to the lengths
of the rods? How is measure of the exterior angle formed by the yellow and blue
rods in the triangle above related to the measures of the angles inside the triangle?
In this lesson you will investigate these questions.
LESSONSTANDARDS
NCTM
OBJECTIVES
LESSON OBJECTIVES
CONTENT
Create a histogram and a stem-and-leaf
PROCESSplot of a data set
Given
a list of data, use a calculator Problem
to graphSolving
a histogram
Number
histograms and stem-and-leaf
plots
Interpret
Reasoning
Algebra
Decide the appropriateness of a histogram and a stem-andleafGeometry
plot for a given data set Communication
Measurement
Connections
Data/Probability
Representation
Investigate inequalities among sides and angles in triangles
Discover the Exterior Angle Conjecture
Practice construction skills
Develop reasoning skills
One Step
To combine the investigations,
pose this problem: “Draw a horizontal line, which we’ll call the
base, and mark two points on it
fairly close together. Make a
triangle that has those two points
as two of its vertices. Now, move
one of the two points along the
base, keeping the opposite side of
the triangle fixed in length. Stop
the movement at various points
and measure all angles and side
lengths. Look for patterns and
make conjectures.” Students might
use geometry software for their
experimentation, or you might
provide sticks or uncooked
spaghetti. While circulating, be
sure some groups focus on exterior angles, some on relative side
and angle measures, and some on
what happens when the triangle
disappears.
LESSON 4.3 Triangle Inequalities
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INTRODUCTION
To help students decide which
rods form a triangle, you might
have them copy to paper, or
imagine copying, three different
rods with the longest rod horizontal and the shorter rods
trying to form the other sides of
the triangle. [Ask] “Under what
conditions can the shorter sides
actually make a triangle?”
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Investigation 1
What Is the Shortest Path
from A to B?
You will need
●
●
a compass
a straightedge
Each person in your group should do each construction. Compare results when
you finish.
Step 1
Given:
A
T
C
T
Construct: CAT
Given:
F
S
H
S
H
F
Step 2 The two shorter
and FH
, are
sides, SH
Step 2
not long enough to
form a triangle with the
.
longest side, FS
Construct: FSH
You should have been able to construct CAT, but not FSH. Why? Discuss
your results with others. State your observations as your next conjecture.
C-20
Triangle Inequality Conjecture
? the length of the
The sum of the lengths of any two sides of a triangle is greater than
third side.
The Triangle Inequality Conjecture relates the lengths of the three sides of a
triangle. You can also think of it in another way: The shortest path between two
points is along the segment connecting them. In other words, the path from A to C
to B can’t be shorter than the path from A to B.
Students can use the Dynamic
Geometry Exploration at
www.keymath.com/DG to help
them understand the Triangle
Inequality Conjecture.
AB = 3 cm
AC + CB = 4 cm
C
A
keymath.com/DG
216
A
C
Guiding Investigation 1
Step 1 The construction can
be done with either a compass
or patty paper. You might want
to have sticks or uncooked
spaghetti available to represent
the line segments. [Alert]
Students may have difficulty
seeing what the constructions
demonstrate about the lengths of
the three sides. [Ask] “What
would happen if the lengths of
the two smaller sides added up to
exactly the same length as the
third side or added up to less
than that of the third side?”
Don’t press students yet to
understand in depth the connection to shortest paths; save that
for Sharing.
Construct a triangle with each set of segments as sides.
CHAPTER 4 Discovering and Proving Triangle Properties
B
AB = 3 cm
AC + CB = 3.1 cm
AB = 3 cm
AC + CB = 3.5 cm
C
A
C
B
A
B A
AB = 3 cm
AC + CB = 3 cm
C
B
[ For an interactive version of this sketch, see the Dynamic Geometry Exploration The Triangle
Inequality at www.keymath.com/DG .]
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Investigation 2
Guiding Investigation 2
Where Are the Largest and Smallest Angles?
Each person should draw a different scalene triangle for
this investigation. Some group members should draw
acute triangles, and some should draw obtuse triangles.
You will need
●
●
a ruler
a protractor
Measure the angles in your triangle. Label the angle
with greatest measure L, the angle with second
greatest measure M, and the smallest angle S.
Step 1
M
S
M
Measure the three sides. Label the longest side l, the
second longest side m, and the shortest side s.
Step 2
Step 3 The largest Step 3
side will be opposite the
largest angle, and so on.
L
L
S
Which side is opposite L? M? S?
Discuss your results with others. Write a conjecture that states where the largest and
smallest angles are in a triangle, in relation to the longest and shortest sides.
C-21
Side-Angle Inequality Conjecture
In a triangle, if one side is longer than another side, then the angle opposite
? . larger than the angle opposite the shorter side
the longer side is Adjacent
interior angle
Exterior
angle
Remote interior angles
So far in this chapter, you have studied interior angles of triangles.
Triangles also have exterior angles. If you extend one side of a
triangle beyond its vertex, then you have constructed an exterior
angle at that vertex.
Each exterior angle of a triangle has an adjacent interior angle
and a pair of remote interior angles. The remote interior angles
are the two angles in the triangle that do not share a vertex with
the exterior angle.
Investigation 3
Exterior Angles of a Triangle
Each person should draw a different scalene triangle for
this investigation. Some group members should draw
acute triangles, and some should draw obtuse triangles.
You will need
●
●
a straightedge
patty paper
Step 1
On your paper, draw a scalene triangle, ABC.
beyond point B and label a point D
Extend AB
. Label the angles as
outside the triangle on AB
shown.
Sharing Ideas (continued)
[Ask] “Can anything be said about the difference in
length of two sides of a triangle?” [The difference
must be less than the length of the third side.]
“Why must the difference be less than the length of
the third side?” [Encourage students to use algebra
to write the claims and see how one follows from
the other: If a b c, then c b a.] [Alert]
If students are confused by the symbols (greater
than) and (less than), review their meaning.
C
c
a
A
b x
B
D
You might also use letters and inequality symbols
in stating the second conjecture: If a b, then
mA mB. The same investigation could also
lead to the converse: If mA mB, then a b.
When phrasing the third conjecture, introduce
the terms adjacent, remote, interior, and exterior.
[ELL] Adjacent means “next to”; remote means “far
away”; interior means “on the inside”; exterior
means “on the outside.”
The wording of this conjecture
may show a lot of variation
among the groups. Encourage
variety and creativity. Groups
may want to reword more than
the end of the conjecture. For
example: “In a triangle, the
smallest angle is opposite the
shortest side and the largest angle
is opposite the longest side.”
Guiding Investigation 3
This investigation may be done
as a follow-along activity.
[Ask] “How could you state
a Triangle Exterior Angle
Inequality Conjecture?” [The
measure of the exterior angle
of a triangle must be greater
than the measure of either
remote interior angle.]
SHARING IDEAS
Have students present a variety
of statements of conjectures.
Lead the class in critiquing them
and in reaching consensus about
which conjecture to use.
When you’ve reached consensus
on the first conjecture, ask how
to explain why the sum of the
lengths of two sides of a triangle
must be greater than the length
of the third. Students will probably keep restating the fact that
you just can’t make a triangle
without that property. Ask about
the relevance of the fact that the
shortest path between two points
is the line segment connecting
them. Help students see that the
shortest-path condition implies
the Triangle Inequality Conjecture. [Ask] “Does the Triangle
Inequality Conjecture imply
the shortest-path condition?”
[It doesn’t, because it makes
no claim that other curves
connecting the two points are
longer than the line segment.]
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Assessing Progress
You can assess students’ skill in
constructing triangles, copying
segments and angles, and
measuring angles. Check their
understanding of various kinds
of triangles.
Step 2
Copy the two remote interior angles, A and C, onto
patty paper to show their sum.
Step 3
How does the sum of a and c compare with x? Use your
patty paper from Step 2 to compare.
Step 4
Discuss your results with your group. State your
observations as a conjecture.
c
a
is equal to the sum of C-22
the measures of the remote
? . interior angles
The measure of an exterior angle of a triangle Triangle Exterior Angle Conjecture
Closing the Lesson
Restate the three conjectures
of this lesson: the Triangle
Inequality Conjecture, the
Side-Angle Inequality Conjecture, and the Triangle Exterior
Angle Conjecture. If you plan to
assign Exercise 11, also mention
that the Triangle Inequality
Conjecture implies that the
difference in length of two sides
of a triangle is less than the
length of the third side; represent
this inequality using letters and
inequality symbols. If many
students seem to be having
difficulty, you may want to
model sample problems, such
as Exercise 5.
BUILDING
UNDERSTANDING
Developing Proof The investigation may have convinced you that the Triangle Exterior
Angle Conjecture is true, but can you explain why it is true for every triangle?
As a group, discuss how to prove the Triangle Exterior Angle Conjecture. Use
reasoning strategies such as draw a labeled diagram, represent a situation
algebraically, and apply previous conjectures. Start by making a diagram and listing
the relationships you already know among the angles in the diagram, then plan out
the logic of your proof.
You will write the paragraph proof of the Triangle Exterior Angle Conjecture in
Exercise 17. EXERCISES
In Exercises 1–4, determine whether it is possible to draw a triangle with sides having
the given measures. If possible, write yes. If not possible, write no and make a sketch
3.
4
5
5
6
demonstrating why it is not possible. 2.
1. 3 cm, 4 cm, 5 cm yes
9
4. 3.5 cm, 4.5 cm, 7 cm
yes
In Exercises 5–10, use your new conjectures to arrange the unknown measures in order
from greatest to least.
a, b, c
5.
6.
The exercises apply the three
conjectures of this lesson. Ask
students which conjectures they
are using to solve each exercise.
3. 5 ft, 6 ft, 12 ft no
7.
c, b, a
c
b
c
2–18 evens
15
Portfolio
14
Journal
17
Group
1–15 odds
Review
19–24
Algebra review 17
|
Helping with the Exercises
70
8.
a
17 in.
68
a
a
9.
a, c, b
a, b, c
w
10.
34
b
b
c
z
72 c
30
a
y
28
v
11. If 54 and 48 are the lengths of two sides of a triangle, what is the range of possible
values for the length of the third side?
6 length 102
Exercise 11 As needed, remind students of the claims
discussed in class: that the sum of the lengths of
the two given sides is greater than the length of the
third side, which is greater than the difference in
the length of the two given sides.
Exercises 1–4 If students are
unsure, encourage them to make
sketches.
218
b, a, c
12 cm
35
a
28 in.
Performance
assessment
c
5 cm
15 in.
Essential
9 cm
b
55
b
ASSIGNING HOMEWORK
12
2. 4 m, 5 m, 9 m no
CHAPTER 4 Discovering and Proving Triangle Properties
x
42
30
v, z, y, w, x
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12. Developing Proof What’s wrong with this
picture? Explain. By the Triangle Inequality
Conjecture, the sum of
11 cm and 25 cm
25 cm should be greater
than 48 cm.
11 cm
13. Developing Proof What’s wrong with this
picture? Explain.
130
b
48 cm
a
In Exercises 14–16, use one of your new conjectures to find the missing measures.
?
14. t p ? 72°
15. r 135°
p
135
125
b 55°, but 55° 130° 180°,
which is impossible by the
Triangle Sum Conjecture.
17. a b c 180° and
x c 180°. Subtract c from
both sides of both equations
to get x 180 c and
a b 180 c. Substitute
a b for 180 c in the first
equation to get x a b.
? 72°
16. x r
t
130
x
58
144
17. Developing Proof Use the Triangle Sum Conjecture to explain
why the Triangle Exterior Angle Conjecture is true. Use the
figure at right.
B
b
18. Read the Recreation Connection below. If you want to know
the perpendicular distance from a landmark to the path of
your boat, what should be the measurement of your bow angle
when you begin recording? 45°
c x
C
a
D
A
Recreation
Geometry is used quite often in sailing. For example, to find the distance
between the boat and a landmark on shore, sailors use a rule called doubling
the angle on the bow. The rule says, measure the angle on the bow (the angle
formed by your path and your line of sight to the landmark, also called your
bearing) at point A. Check your bearing until, at point B, the bearing is double
the reading at point A. The distance traveled from A to B is also the distance
from the landmark to your new position.
L
A
B
LESSON 4.3 Triangle Inequalities
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21. By the Triangle Sum
Conjecture, the third angle
must measure 36° in the small
triangle, but it measures 32° in
the large triangle. These are the
same angle, so they can’t have
different measures.
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Review
In Exercises 19 and 20, calculate each lettered angle measure.
19.
21. What’s wrong with this
picture of TRG? Explain.
20.
d
c
f
e
32
EXTENSIONS
A. Ask students to use geometry
software to explore congruence
shortcuts. It’s especially useful for
the one-step investigation.
a
d
T
g
c
b
a
h
b
a 90°,
b 68°,
22 c 112°, d 112°,
e 68°, f 56°,
g 124°, h 124°
38
72
L
72
74
N
74
R
G
a 52°, b 38°, c 110°, d 35°
In Exercises 22–24, complete the statement of congruence.
B. Use Take Another Look
activity 4 on page 255.
?
3.6 22. BAR ? FNK
4.2 23. FAR ABE
HJ
3.6 24. HG
cannot be
? determined
HEJ N
E
A
A
J
52
B
R
G
K
38
R
F
E
H
O
RANDOM TRIANGLES
Imagine you cut a 20 cm straw in two randomly selected places anywhere along
You can use Fathom to
generate many sets of
random numbers quickly.
You can also set up tables
to view your data, and
enter formulas to calculate
quantities based on your
data.
its length. What is the probability that the three pieces will form a triangle? How
do the locations of the cuts affect whether or not the pieces will form a triangle?
Explore this situation by cutting a straw in different ways, or use geometry
software to model different possibilities. Based on your informal exploration,
predict the probability of the pieces forming a triangle.
Now generate a large number of randomly chosen lengths to simulate the cutting of
the straw. Analyze the results and calculate the probability based on your data. How
close was your prediction?
Your project should include
Supporting the
OUTCOMES
After students have worked on the project,
discuss how collecting more and more
data (or pooling data) gets you closer and
closer to the theoretical probability. Go
to www.keymath.com/DG for a Fathom
demonstration. (To avoid wasting straws, ask
students to randomly bend pipe cleaners.)
220
Your prediction and an explanation of how you arrived at it.
Your randomly generated data.
An analysis of the results and your calculated probability.
An explanation of how the location of the cuts affects the chances of a triangle
being formed.
Presentation of data is organized and clear.
Explanations of predictions and
descriptions of the results are consistent.
If lengths are generated using a graphing
calculator, Fathom, or another randomlength generator, the experimental
probability for large samples will be
around 25%.
CHAPTER 4 Discovering and Proving Triangle Properties
A graph of the
sample space
uses shading to
show cut
combinations
that do
produce a
triangle.
y
1
1/2
0
1/2
1
x