LESSON OBJECTIVES

� Create a histogram and a stem-and-leaf plot of a data set� Given a list of data, use a calculator to graph a histogram� Interpret histograms and stem-and-leaf plots� Decide the appropriateness of a histogram and a stem-and-

leaf plot for a given data set

LESSON 4.3 Triangle Inequalities 215

L E S S O N

4.3Readers are plentiful, thinkers

are rare.

HARRIET MARTINEAU

Triangle InequalitiesHow long must each side of this drawbridge be so that the bridge spans the river

when both sides come down?

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?

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.

Drawbridges over theChicago River inChicago, Illinois

L E S S O N

4.3

LESSON OBJECTIVES

� Investigate inequalities among sides and angles in triangles� Discover the Exterior Angle Conjecture� Practice construction skills � Develop reasoning skills

NCTM STANDARDS

CONTENT PROCESS

Number Problem Solving

� Algebra � Reasoning

� Geometry � Communication

� Measurement � Connections

Data/Probability Representation

YO

PLANNING

LESSON OUTLINE

One day:25 min Investigation

10 min Sharing

5 min Closing

5 min Exercises

MATERIALS

� construction tools� protractors� sticks or uncooked spaghetti, optional� Triangle Exterior Angle

Conjecture (W), optional� Sketchpad activity Triangle

Inequalities, optional

TEACHING

This lesson concerns three prop-

erties of triangles: the triangle in-

equality, the side-angle inequality,

and the exterior angle property.

One StepTo combine the investigations,

pose this problem: “Draw a hori-

zontal 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 exte-

rior angles, some on relative side

and angle measures, and some on

what happens when the triangle

disappears.

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Each person in your group should do each construction. Compare results when

you finish.

Step 1 Construct a triangle with each set of segments as sides.

Given:

Construct: �CAT

Given:

Construct: �FSH

Step 2 You should have been able to construct �CAT, but not �FSH. Why? Discuss

your results with others. State your observations as your next conjecture.

F

S

F

S

H

H

C

A

C

A

T

T

Triangle Inequality Conjecture

The sum of the lengths of any two sides of a triangle is �? the length of the

third side.

C-20

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.

AB = 3 cmAC + CB = 4 cm

AB = 3 cmAC + CB = 3.5 cm

A B A B

CC

AB = 3 cmAC + CB = 3 cm

A BC

AB = 3 cmAC + CB = 3.1 cm

A B

C

Investigation 1What Is the Shortest Path from A to B?You will need

● a compass

● a straightedge

[� For an interactive version of this sketch, see the Dynamic Geometry Exploration The TriangleInequality at .�]www.keymath.com/DGkeymath.com/DG

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 hori-

zontal 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?”

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 connec-

tion to shortest paths; save that

for Sharing.

Students can use the Dynamic

Geometry Exploration at

www.keymath.com/DG to help

them understand the Triangle

Inequality Conjecture.

216 CHAPTER 4 Discovering and Proving Triangle Properties

greater than

Step 2 The two shortersides, SH� and FH�, arenot long enough toform a triangle with thelongest side, FS�.

DG4TE_883_04.qxd 10/24/06 6:38 PM Page 216

Investigation 3Exterior Angles of a TriangleEach person should draw a different scalene triangle for

this investigation. Some group members should draw

acute triangles, and some should draw obtuse triangles.

Step 1 On your paper, draw a scalene triangle, �ABC.

Extend AB� beyond point B and label a point D

outside the triangle on AB��. Label the angles as

shown.

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.

Each person should draw a different scalene triangle for

this investigation. Some group members should draw

acute triangles, and some should draw obtuse triangles.

Step 1 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 2 Measure the three sides. Label the longest side l, the

second longest side m, and the shortest side s.

Step 3 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.

L

M SM

L S

Side-Angle Inequality Conjecture

In a triangle, if one side is longer than another side, then the angle opposite

the longer side is �? .

C-21

Adjacentinterior angleExterior

angle

Remote interior angles

Investigation 2Where Are the Largest and Smallest Angles?

You will need

● a ruler

● a protractor

You will need

● a straightedge

● patty paper

Aa

c

b xDB

C

Guiding Investigation 2

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 prob-

ably 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 Conjec-

ture. [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.]

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.

You might also use letters and inequality symbols

in stating the second conjecture: If a � b, then

m�A � m�B. The same investigation could also

lead to the converse: If m�A � m�B, 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.”

LESSON 4.3 Triangle Inequalities 217

Step 3 The largest side will be opposite thelargest angle, and so on.

larger than the angle opposite the shorter side

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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.

Triangle Exterior Angle Conjecture

The measure of an exterior angle of a triangle �? .

C-22

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. �

EXERCISESIn 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

demonstrating why it is not possible.

1. 3 cm, 4 cm, 5 cm 2. 4 m, 5 m, 9 m 3. 5 ft, 6 ft, 12 ft 4. 3.5 cm, 4.5 cm, 7 cm

In Exercises 5–10, use your new conjectures to arrange the unknown measures in order

from greatest to least.

5. 6. 7.

8. 9. 10.

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?

30�a

c

b

72�

55�

68�

b c

a

c

b

a70�

35�

�

ca

17 in.15 in.

28 in.

a

b

c 30�

42�

34�

28�

z

v

y

w

x

b

a

c9 cm

5 cm12 cm

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.

218 CHAPTER 4 Discovering and Proving Triangle Properties

Assessing ProgressYou can assess students’ skill in

constructing triangles, copying

segments and angles, and

measuring angles. Check their

understanding of various kinds

of triangles.

Closing the Lesson

Restate the three conjectures

of this lesson: the Triangle

Inequality Conjecture, the

Side-Angle Inequality Conjec-

ture, 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.

BUILDINGUNDERSTANDING

The exercises apply the three

conjectures of this lesson. Ask

students which conjectures they

are using to solve each exercise.

ASSIGNING HOMEWORK

Essential 2–18 evens

Performanceassessment 15

Portfolio 14

Journal 17

Group 1–15 odds

Review 19–24

Algebra review 17

|

� Helping with the Exercises

Exercises 1–4 If students are

unsure, encourage them to make

sketches.

is equal to the sum ofthe measures of the remoteinterior angles

yes

2. 3.

no

6 � length � 102

noyes

a, b, c c, b, a

b, a, c

a, c, b a, b, c

v, z, y, w, x

4 5

9

5 6

12

DG4TE_883_04.qxd 10/24/06 6:38 PM Page 218

12. Developing Proof What’s wrong with this 13. Developing Proof What’s wrong with this

picture? Explain. picture? Explain.

In Exercises 14–16, use one of your new conjectures to find the missing measures.

14. t � p � �? 15. r � �? 16. x � �?

17. Developing Proof Use the Triangle Sum Conjecture to explain

why the Triangle Exterior Angle Conjecture is true. Use the

figure at right.

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?

x

144�

r

58�130�

p

t 135�

130�

a

b125�

11 cm

48 cm

25 cm

Recreation

Geometry is used quite often in sailing. For example, to find the distancebetween the boat and a landmark on shore, sailors use a rule called doublingthe angle on the bow. The rule says, measure the angle on the bow (the angleformed by your path and your line of sight to the landmark, also called yourbearing) at point A. Check your bearing until, at point B, the bearing is doublethe reading at point A. The distance traveled from A to B is also the distancefrom the landmark to your new position.

A B

L

A

C

B

b

ac x

D

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.

LESSON 4.3 Triangle Inequalities 219

By the Triangle Inequality Conjecture, the sum of

11 cm and 25 cm should be greater

than 48 cm.

135° 72° 72°

45°

b � 55°, but 55° � 130° � 180°,which is impossible by theTriangle Sum Conjecture.

DG4TE_883_04.qxd 10/24/06 6:38 PM Page 219

Review

In Exercises 19 and 20, calculate each lettered angle measure.

19. 20.

In Exercises 22–24, complete the statement of congruence.

22. �BAR � � �? 23. �FAR � � �? 24. HG�� � HJ��HEJ � � �?

E O

G

J

H

52� 38�

R

A

NK

FB R

E A

ed

hb

ca

22�

f

g

a b

38�

32�

c

d

�

RANDOM TRIANGLES

Imagine you cut a 20 cm straw in two randomly selected places anywhere along

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

� 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.

You can use Fathom togenerate many sets ofrandom numbers quickly.You can also set up tablesto view your data, andenter formulas to calculatequantities based on yourdata.

21. What’s wrong with this

picture of �TRG? Explain.

74� 74�

72� 72�

T

NL

R G

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.

EXTENSIONS

A. Ask students to use geometry

software to explore congruence

shortcuts. It’s especially useful for

the one-step investigation.

B. Use Take Another Look

activity 4 on page 255.

220 CHAPTER 4 Discovering and Proving Triangle Properties

3.6 4.2 3.6

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.)

� 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 random-

length generator, the experimental

probability for large samples will be

around 25%.

� A graph of the

sample space

uses shading to

show cut

combinations

that do

produce a

triangle.

OUTCOMESSupporting the

x

y

1

11/20

1/2

a � 52°, b � 38°, c � 110°, d � 35°

ABE FNK cannot bedetermined

a � 90°,b � 68°,

c � 112°, d � 112°,e � 68°, f � 56°,

g � 124°, h � 124°

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