Post on 19-May-2018
© Carnegie Learning
451
5Properties of Triangles
5.1 Name That Triangle!
Classifying Triangles on the Coordinate Plane . . . . . . . . 453
5.2 Inside Out
Triangle Sum, Exterior Angle, and Exterior
Angle Inequality Theorems . . . . . . . . . . . . . . . . . . . . . . . 461
5.3 Trade Routes and Pasta Anyone?
The Triangle Inequality Theorem . . . . . . . . . . . . . . . . . . . 479
5.4 Stamps Around the World
Properties of a 458–458–908 Triangle . . . . . . . . . . . . . . . . 489
5.5 More Stamps, Really?
Properties of a 30°–60°–90° Triangle . . . . . . . . . . . . . . . 497
A lot of people use email
but there is still a need to “snail” mail too. Mail isn’t really delivered by snails—it’s just a
comment on how slow it is compared to a computer.
© Carnegie Learning
451A Chapter 5 Properties of Triangles
5
Chapter 5 Overview
This chapter focuses on properties of triangles, beginning with classifying triangles on the coordinate plane. Theorems
involving angles and side lengths of triangles are presented. The last two lessons discuss properties and theorems of
45º-45º-90º triangles and 30º-60º-90º triangles.
Lesson TEKS Pacing Highlights
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5.1
Classifying
Triangles on the
Coordinate
Plane
2.B 1
This lesson provides opportunities for
students to graph and classify triangles on
the coordinate plane by their side lengths
and angle measures.
As a culminating activity, students
classify India’s Golden Triangle given the
cities’ coordinates.
X X
5.2
Triangle Sum,
Exterior Angle,
and Exterior
Angle Inequality
Theorems
6.D 1
In this lesson, students prove the Triangle
Sum Theorem, Exterior Angle Theorem, and
Exterior Angle Inequality Theorem.
Questions ask students to investigate the
side lengths and angle measures of triangles
before proving the theorems.
X X
5.3
The Triangle
Inequality
Theorem
5.D 1
Students complete an activity with pasta
strands to investigate the possible side
lengths that can form triangles. Students
then prove the Triangle Inequality Theorem.
X X
5.4
Properties of a
45°–45°–90°
Triangle
7.A
9.B1
Students investigate the properties of
45°245°290° triangles in this lesson.
Questions ask students to apply the
45°245°290° Triangle Theorem and
construction to solve problems and verify
properties of 45°245°290° triangles.
X
5.5
Properties of a
30°–60°–90°
Triangle
9.B 1
Students investigate the properties of
30°-60°-90° triangles in this lesson.
Questions ask students to apply the
30°260°290° Triangle Theorem and
construction to solve problems and verify
properties of 30°260°290° triangles.
As a culminating activity, students compare
the properties of 45°245°290° triangles
with 30°260°290° triangles.
X X
© Carnegie Learning
5
Chapter 5 Properties of Triangles 451B
Skills Practice Correlation for Chapter 5
Lesson Problem Set Objectives
5.1
Classifying Triangles on the Coordinate Plane
1 – 6Determine the possible locations of a point to create triangles on the coordinate plane given a line segment
7 – 12Graph triangles on the coordinate plane given vertex coordinates and classify the triangles based on the side lengths
13 – 18Graph triangles on the coordinate plane given vertex coordinates and classify the triangles based on the angle measures
5.2
Triangle Sum, Exterior Angle, and Exterior
Angle Inequality Theorems
Vocabulary
1 – 6 Determine the measure of missing angle measures in triangles
7 – 12 Determine the order of side lengths given information in diagrams
13 – 18 Identify interior, exterior, and remote interior angles of triangles
19 – 24 Solve for x given triangle diagrams
25 – 30Write two inequalities needed to prove the Exterior Angle Inequality Theorem given triangle diagrams
5.3The Triangle Inequality Theorem
Vocabulary
1 – 6 Order angle measures of triangles without measuring
7 – 16 Determine whether it is possible to form a triangle from given side lengths
17 – 22 Write inequalities to describe possible unknown side lengths of triangles
5.4Properties of a 45°–45°–90° Triangle
Vocabulary
1 – 4 Determine the length of the hypotenuse of 45°245°290° triangles
5 – 8 Determine the lengths of the legs of 45°245°290° triangles
9 – 12 Solve problems involving 45°245°290° triangles
13 – 16 Determine the area of 45°245°290° triangles
17 – 20 Solve problems involving 45°245°290° triangles
21 – 24 Construct 45°245°290° triangles
5.5Properties of a 30°–60°–90° Triangle
Vocabulary
1 – 4 Determine the measure of indicated interior angles
5 – 8Determine the length of the long leg and the hypotenuse of 30°260°290° triangles
9 – 12 Determine the lengths of the legs of 30°260°290° triangles
13 – 16Determine the length of the short leg and the hypotenuse of 30°260°290° triangles
17 – 20 Determine the area of 30°260°290° triangles
21 – 24 Construct 30°260°290° triangles
© Carnegie Learning
452 Chapter 5 Properties of Triangles
5
© Carnegie Learning
453A
ESSENTIAL IDEAS
Given the coordinates of two points, a third
point is located to form an equilateral
triangle, an isosceles triangle, a scalene
triangle, an acute triangle, an obtuse
triangle, and a right triangle.
Given the coordinates of three points,
algebra is used to describe characteristics
of the triangle.
TEXAS ESSENTIAL KNOWLEDGE
AND SKILLS FOR MATHEMATICS
(2) Coordinate and transformational geometry.
The student uses the process skills to understand
the connections between algebra and geometry
and uses the one- and two-dimensional
coordinate systems to verify geometric
conjectures. The student is expected to:
(B) derive and use the distance, slope, and
midpoint formulas to verify geometric
relationships, including congruence
of segments and parallelism or
perpendicularity of pairs of lines
5.1Name That Triangle!Classifying Triangles on the Coordinate Plane
LEARNING GOALS
In this lesson, you will:
Determine the coordinates of a third vertex of a triangle, given the coordinates
of two vertices and a description of the triangle.
Classify a triangle given the locations of its vertices on a coordinate plane.
© Carnegie Learning
453B Chapter 5 Properties of Triangles
5
Overview
Students are given coordinates for 2 points and will determine a third set of coordinates that satisfy a
speci%c triangle description. Next, students are given the coordinates of three vertices of different
triangles and will describe the triangle using side lengths and angle measurements. Using a map,
students transfer a location onto a coordinate plane and answer questions related to the situation.
© Carnegie Learning
5.1 Classifying Triangles on the Coordinate Plane 453C
5
Warm Up
The coordinates of two points A (26, 210) and B (4, 210) are given.
A (26, 210) B (4, 210)
C (26, y) C9 (4, y)
Describe all possible locations for the coordinates of point C such that triangle ABC is a right triangle.
Point C could have the coordinates (26, y) or (4, y), where y is any real number.
© Carnegie Learning
453D Chapter 5 Properties of Triangles
5
© Carnegie Learning
5.1 Classifying Triangles on the Coordinate Plane 453
5
453
5.1
Because you may soon be behind the steering wheel of a car, it is important to
know the meaning of the many signs you will come across on the road. One of the
most basic is the yield sign. This sign indicates that a driver must prepare to stop to
give a driver on an adjacent road the right of way. The first yield sign was installed in
the United States in 1950 in Tulsa, Oklahoma, and was designed by a police officer of
the town. Originally, it was shaped like a keystone, but over time, it was changed.
Today, it is an equilateral triangle and is used just about everywhere in the world.
Although some countries may use different colors or wording (some countries call it a
“give way” sign), the signs are all the same in size and shape.
Why do you think road signs tend to be different, but basic, shapes, such as
rectangles, triangles, and circles? Would it matter if a stop sign was an irregular
heptagon? Does the shape of a sign make it any easier or harder to recognize?
LEARNING GOALS
In this lesson, you will:
Determine the coordinates of a third vertex of a triangle, given the coordinates
of two vertices and a description of the triangle.
Classify a triangle given the locations of its vertices on a coordinate plane.
Name That Triangle!Classifying Triangles on the Coordinate Plane
Students may need a
reminder of the three
classi%cations of triangles
by side, scalene,
isosceles, and equilateral,
and of the three
classi%cations by angle,
acute, obtuse, and right.
Help students create a
graphic organizer where
they write the type in
the %rst column, sketch
the type in the second
column, and description
in their own words in the
last column.
© Carnegie Learning
454 Chapter 5 Properties of Triangles
5
Problem 1
Students are given two points
on the coordinate plane and will
determine all possible locations
for a third point that meet
speci%c triangular constraints
related to side lengths and
angle measures.
Grouping
Have students complete
Questions 1 through 3 with a
partner. Then have students
share their responses as
a class.
Guiding Questions for Share Phase, Questions 1 through 3
How can perpendicular
bisectors help determine
the possible locations for
point C?
How can the perpendicular
bisector of line segment AB
be helpful?
Is it possible for point C
to have an in%nite number
of locations to satisfy
this constraint?
How many possible locations
for point C are there if
triangle ABC is equiangular?
PROBLEM 1 Location, Location, Location!
1. The graph shows line segment AB with endpoints at A (26, 7) and B (26, 3).
Line segment AB is a radius for congruent circles A and B.
2 4 6x
28210 26 24 22
2
0
6
4
8
y
A
B
10
26
24
22
2. Using ___
AB as one side of a triangle, determine a
location for point C on circle A or on circle B such that
triangle ABC is:
a. a right triangle.
Point C can have an infinite number of locations
as long as the location satisfies one of the
following conditions:
Point C is located at the point (210, 7) or
(22, 7) on Circle A.
Point C is located at the point (210, 3) or (22, 3)
on Circle B.
b. an acute triangle.
Point C can have an infinite number of locations as
long as the location satisfies one of the following conditions:
Point C is located anywhere on circle A between the y-values of
3 and 7, except where x 5 26.
Point C is located anywhere on circle B between the y-values of
3 and 7, except where x 5 26.
If you are unsure
about where this point would lie, think about the steps it took to construct different triangles. Draw
additional lines or figures on your coordinate plane to
help you.
© Carnegie Learning
5.1 Classifying Triangles on the Coordinate Plane 455
5
c. an obtuse triangle.
Point C can have an infinite number of locations as long as the location satisfies
one of the following conditions:
Point C is located at any point on circle A with a y-value greater than
7, except where x 5 26.
Point C is located at any point on circle B with a y-value less than
3, except where x 5 26.
3. Using ___
AB as one side of a triangle, determine the location for point C on circle A or on
circle B such that triangle ABC is:
a. an equilateral triangle.
Point C can have two possible locations. Circle A and circle B intersect at two
locations. Either point of intersection is a possible location for point C.
b. an isosceles triangle.
Point C can have an infinite number of locations as long as the location satisfies
one of the following conditions:
Point C is located anywhere on circle A, except where x 5 26.
Point C is located anywhere on circle B, except where x 5 26.
c. a scalene triangle.
Point C can have an infinite number of locations as long as the location is not at
any of the locations mentioned in parts (a) or (b).
© Carnegie Learning
456 Chapter 5 Properties of Triangles
5
Problem 2
Students will graph three
points and use algebra to
determine the characteristics
of the triangle with respect to
the length of its sides and the
measures of the angles. They
use the Distance Formula to
classify the triangle as scalene,
isosceles, or equilateral. Next,
the slope formula and the
Pythagorean Theorem are used
to classify a triangle as a right
triangle. The second activity is
similar to the %rst activity.
Grouping
Have students complete
Questions 1 and 2 with a
partner. Then have students
share their responses as a class.
Guiding Questions for Share Phase, Questions 1 and 2
What formulas are used to
determine the length of the
sides of the triangle?
What formula helps to
determine if the triangle
contains a right angle?
What is the relationship
between the slopes of
perpendicular lines?
PROBLEM 2 What’s Your Name Again?
1. Graph triangle ABC using points A (0, 24), B (0, 29), and C (22, 25).
28 26 24 22
22
24
26
20 4 6 8x
28
y
8
6
4
2
C
A
B
2. Classify triangle ABC.
a. Determine if triangle ABC is scalene, isosceles, or equilateral.
Explain your reasoning.
Because line segment AB is vertical, I can subtract the y-coordinates of the
endpoints to determine its length.
AB 5 24 2 (29)
5 5
BC 5 √_______________________
(22 2 0)2 1 (25 2 (29))2
5 √___________
(22)2 1 (4)2
5 √_______
4 1 16
5 √___
20
AC 5 √_______________________
(22 2 0)2 1 (25 2 (24))2
5 √_____________
(22)2 1 (21)2
5 √______
4 1 1
5 √__
5
Triangle ABC is scalene because no two side lengths
are equal.
These classifications are all about the
lengths of the sides. How can I determine the lengths of the sides of
this triangle?
© Carnegie Learning
5.1 Classifying Triangles on the Coordinate Plane 457
5
?
b. Explain why triangle ABC is a right triangle.
Line segment AB is a vertical line on the y-axis. This means the slope is
undefined.
Slope of ___
AC :
m 5 y
2 2 y
1 _______ x2 2 x
1
m 5 25 2 (24)
__________
22 2 0 5 21 ___
22 5 1 __
2
Slope of ___
BC :
m 5 y
2 2 y
1 _______ x2 2 x
1
m 5 25 2 (29)
__________
22 2 0 5 4
___ 22
5 22
Triangle ABC is a right triangle. The slopes of the segments that form angle C are
negative reciprocals of each other, so they must be perpendicular, which means
they form a right angle.
c. Zach does not like using the slope formula. Instead, he decides to use the
Pythagorean Theorem to determine if triangle ABC is a right triangle because
he already determined the lengths of the sides. His work is shown.
Zach
a2 1 b2 5 c2
( √__ 5 ) 2 1 ( √
___ 20 ) 2 5 5 2
5 1 20 5 25
25 5 25
He determines that triangle ABC must be a right triangle because the sides satisfy
the Pythagorean Theorem. Is Zach’s reasoning correct? Explain why or why not.
Yes. Zach’s reasoning is correct. The Pythagorean Theorem only holds true for
right triangles. Because the side lengths of triangle ABC satisfy the
Pythagorean Theorem, Zach proved that triangle ABC is a right triangle.
© Carnegie Learning
458 Chapter 5 Properties of Triangles
5
Grouping
Have students complete
Questions 3 and 4 with a
partner. Then have students
share their responses as a class.
Guiding Questions for Share Phase, Questions 3 and 4
What formulas are used to
determine the length of the
sides of the triangle?
What formula helps to
determine if triangle ABC
contains a right angle?
If triangle ABC is not a
right triangle, what are the
other possibilities?
3. Graph triangle ABC using points A (22, 4), B (8, 4), and C (6, 22).
28 26 24 22
22
24
26
20 4 6 8x
28
y
8
6
4
2
C
A B
4. Classify triangle ABC.
a. Determine if triangle ABC is a scalene, an isosceles, or an equilateral triangle.
Explain your reasoning.
Line segment AB is horizontal so I can determine its length by subtracting the
x-coordinates of its endpoints.
AB 5 8 2 (22)
5 10
BC 5 √___________________
(6 2 8)2 1 (22 2 4)2
5 √_____________
(22)2 1 (26)2
5 √_______
4 1 36
5 √___
40
AC 5 √_____________________
(6 2 (22))2 1 (22 2 4)2
5 √___________
(8)2 1 (26)2
5 √________
64 1 36
5 √____
100
5 10
Because sides AB and AC are equal, triangle ABC must be isosceles. The triangle
is not equilateral, though, because the length of the third side, BC, is not equal to
the other two lengths.
© Carnegie Learning
5.1 Classifying Triangles on the Coordinate Plane 459
5
b. Determine if triangle ABC is a right triangle. Explain your reasoning. If it is not a right
triangle, use a protractor to determine what type of triangle it is.
Line segment AB is a horizontal line, so the slope is 0.
Slope of line segment BC:
m 5 y
2 2 y
1 _______ x2 2 x
1
m 5 4 2 (22)
________ 8 2 6
5 6 __ 2 5 3
Slope of line segment AC:
m 5 y
2 2 y
1 _______ x2 2 x
1
m 5 4 2 (22)
________ 22 2 6
5 6 ___ 28
5 2 3
__ 4
Triangle ABC is not a right triangle because none of the line segments has a
perpendicular relationship with another line segment.
/A 5 30°
/B 5 70°
/C 5 80°
Because all three measures have measures that are less than 90°, triangle ABC must
be an acute triangle.
© Carnegie Learning
460 Chapter 5 Properties of Triangles
5
Problem 3
Students use a map to
determine approximate
coordinates of three
destinations. They connect the
locations to form a triangle and
classify the triangle.
Grouping
Have students complete
Question 1 with a partner.
Then have students share their
responses as a class.
Guiding Questions for Share Phase, Question 1
How is using the origin as
a location for one of the
cities helpful?
Which city did you graph
%rst? Why?
Can you locate a second city
on the x-axis? Which city?
Is the third city located above
or below the x-axis? Why?
How did you determine the
location of the third city?
PROBLEM 3 India’s Golden Triangle
1. India’s Golden Triangle is a very popular tourist destination. The vertices of the triangle
are the three historical cities of Delhi, Agra (Taj Mahal), and Jaipur.
The locations of these three cities can be represented on the coordinate plane
as shown.
x
y
(0, 0) (134, 0)
(100, 105)
Rohtak
Alwar
Fatehpur Sikri
Delhi
Jaipur Agra
Classify India’s Golden Triangle.
JD 5 √_____________________
(100 2 0)2 1 (105 2 0)2 DA 5 √________________________
(100 2 134)2 1 (105 2 0)2
5 √___________
1002 1 1052 5 √_____________
(234)2 1 1052
5 √________________
10,000 1 11,025 5 √______________
1156 1 11,025
5 √_______
21,025 5 √_______
12,181
5 145 ¯ 110.37
Line segment JA is horizontal so I can determine its length by subtracting the
x-coordinates of its endpoints.
JA 5 134 2 0
5 134
India’s Golden Triangle is an acute scalene triangle because each side is a different
length and each angle is less than 90 degrees.
Be prepared to share your solutions and methods.
© Carnegie Learning
5.1 Classifying Triangles on the Coordinate Plane 460A
5
Check for Students’ Understanding
The coordinates of two points A (23, 5) and B (4, 10) are given.
A (23, 5)
B (4, 10)C9 (23, 10)
C (4, 5)
Describe all possible locations of point C such that triangle ABC is a right triangle.
C could have the coordinates (23, 10).
C could have the coordinates (4, 5).
C could be anywhere on a line described by the equation y 5 2 7
__
5 x 1
78 ___
5 .
C could be anywhere on a line described by the equation y 5 2 7
__
5 x 1 4 __
5 .
m 5 y
1 2 y
2 _______
x1 2 x
2
m 5 10 2 5 _______ 4 2 23
5 5 __ 7
(y 2 10) 5 2 7 __
5 (x 2 4)
y 5 2 7 __
5 x 1
78 ___
5
(y 2 5) 5 2 7
__
5 (x 2 23)
y 5 2 7 __
5 x 1 4 __
5
© Carnegie Learning
460B Chapter 5 Properties of Triangles
5
© Carnegie Learning
461A
Inside OutTriangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems
5.2
ESSENTIAL IDEAS
The Triangle Sum Theorem states: “The sum
of the measures of the interior angles of a
triangle is 180°.”
The longest side of a triangle lies opposite
the largest interior angle.
The shortest side of a triangle lies opposite
the smallest interior angle.
The remote interior angles of a triangle are
the two interior angles non-adjacent to the
exterior angle.
The Exterior Angle Theorem states: “The
measure of the exterior angle of a triangle is
equal to the sum of the measures of the two
remote interior angles of the triangle.”
The Exterior Angle Inequality Theorem
states: “An exterior angle of a triangle is
greater than either of the remote interior
angles of the triangle.”
TEXAS ESSENTIAL KNOWLEDGE
AND SKILLS FOR MATHEMATICS
(6) Proof and congruence. The student uses
the process skills with deductive reasoning to
prove and apply theorems by using a variety of
methods such as coordinate, transformational,
and axiomatic and formats such as two-column,
paragraph, and &ow chart. The student is
expected to:
(D) verify theorems about the relationships
in triangles, including proof of the
Pythagorean Theorem, the sum of interior
angles, base angles of isosceles triangles,
midsegments, and medians, and apply
these relationships to solve problems
LEARNING GOALS KEY TERMS
Triangle Sum Theorem
remote interior angles of a triangle
Exterior Angle Theorem
Exterior Angle Inequality Theorem
In this lesson, you will:
Prove the Triangle Sum Theorem.
Explore the relationship between the interior
angle measures and the side lengths of
a triangle.
Identify the remote interior angles of
a triangle.
Identify the exterior angle of a triangle.
Explore the relationship between the
exterior angle measure and two remote
interior angles of a triangle.
Prove the Exterior Angle Theorem.
Prove the Exterior Angle
Inequality Theorem.
© Carnegie Learning
461B Chapter 5 Properties of Triangles
5
Overview
Students informally show and formally prove the Triangle Sum Theorem. Next, students explore the
effect the angle measure has on the length of the side opposite the angle in a triangle. As the angle
measure increases, the length of the side opposite the angle increases. As the angle measure decreases,
the length of the side opposite the angle decreases. These relationships are the foundation of the Hinge
Theorem introduced in a later chapter. Students prove the Exterior Angle Theorem and the Exterior Angle
Inequality Theorem. Maps are used to model triangles, and students answer questions related to the
problem situations.
© Carnegie Learning
5
5.2 Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems 461C
Warm Up
Triangle RTG is shown.
R
T
G
1. If we increase m/T in triangle RTG, what effect will this have on m/R and m/G?
If m/T increases in triangle RTG, m/R or m/G will decrease because the sum of the measures
of the interior angles of a triangle is 180°.
2. If we decrease m/T in triangle RTG, what effect will this have on m/R and m/G?
If m/T decreases in triangle RTG, m/R or m/G will increase because the sum of the measures
of the interior angles of a triangle is 180°.
3. If we increase the length of side GT in triangle RTG, what effect will this have on the lengths of sides
RT and RG?
If the length of side GT increases in triangle RTG, RT or RG will increase because a triangle is a
closed figure.
4. If we decrease the length of side GT in triangle RTG, what effect will this have on the lengths of
sides RT and RG?
If the length of side GT decreases in triangle RTG, RT or RG will decrease because a triangle is a
closed figure.
© Carnegie Learning
461D Chapter 5 Properties of Triangles
5
© Carnegie Learning
5
5.2 Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems 461
461
LEARNING GOALS KEY TERMS
Triangle Sum Theorem
remote interior angles of a triangle
Exterior Angle Theorem
Exterior Angle Inequality Theorem
In this lesson, you will:
Prove the Triangle Sum Theorem.
Explore the relationship between the interior angle measures and the side lengths of a triangle.
Identify the remote interior angles of a triangle.
Identify the exterior angle of a triangle.
Explore the relationship between the exterior angle measure and two remote interior angles of a triangle.
Prove the Exterior Angle Theorem.
Prove the Exterior Angle Inequality Theorem.
Easter Island is one of the remotest islands on planet Earth. It is located in the
southern Pacific Ocean approximately 2300 miles west of the coast of Chile. It
was discovered by a Dutch captain in 1722 on Easter Day. When discovered, this
island had few inhabitants other than 877 giant statues, which had been carved out of
rock from the top edge of a wall of the island’s volcano. Each statue weighs several
tons, and some are more than 30 feet tall.
Several questions remain unanswered and are considered mysteries. Who built these
statues? Did the statues serve a purpose? How were the statues transported on
the island?
Inside OutTriangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems
5.2
© Carnegie Learning
462 Chapter 5 Properties of Triangles
5
Problem 1
First students manipulate the
three interior angles of a triangle
to informally show they form
a line, thus showing the sum
of the measures of the three
interior angles of a triangle is
180 degrees. Then students will
use a two-column proof to prove
the Triangle Sum Theorem.
Grouping
Have students complete
Questions 1 and 2 with a
partner. Then have students
share their responses as
a class.
Guiding Questions for Share Phase, Questions 1 and 2
What is a straight angle?
What is the measure of a
straight angle?
Does a line have a
degree measure?
If three angles form a line
when arranged in an adjacent
con%guration, what is the
sum of the measures of the
three angles?
What do you know about
the sum of the measures of
angles 3, 4, and 5?
What is the relationship
between angle 1 and angle 4?
What is the relationship
between angle 2 and angle 5?
What do you know about
the sum of the measures of
angles 1, 2, and 3?
What properties are used to
prove this theorem?
How many steps is your proof?
Do your classmates have the same number of steps?
Did your classmates use different properties to prove this theorem?
PROBLEM 1 Triangle Interior Angle Sums
1. Draw any triangle on a piece of paper.
Tear off the triangle’s three angles. Arrange
the angles so that they are adjacent angles.
What do you notice about the sum of these
three angles?
The sum of the angles is 180° because they
form a straight line.
2. How could you use constructions to determine
the sum of the angles of a triangle?
I could construct a triangle, duplicate the 3 angles, and position them as adjacent angles.
The Triangle Sum Theorem states: “the sum of the measures of the interior angles of a
triangle is 180°.”
3. Prove the Triangle Sum Theorem using the diagram shown.
C D
A B
4 53
21
Given: Triangle ABC with ___
AB || ___
CD
Prove: m/1 1 m/2 1 m/3 5 180°
Statements Reasons
1. Triangle ABC with ‹
___ › AB ||
‹
___ › CD 1. Given
2. m/4 1 m/3 1 m/5 5 180° 2. Angle Addition Postulate and
Definition of straight angle
3. /1 ˘ /4 3. Alternate Interior Angle Theorem
4. m/1 5 m/4 4. Definition of congruent angles
5. /2 ˘ /5 5. Alternate Interior Angle Theorem
6. m/2 5 m/5 6. Definition of congruent angles
7. m/1 1 m/3 1 m/2 5 180° 7. Substitution Property of Equality
8. m/1 1 m/2 1 m/3 5 180° 8. Associative Property of Addition
Think about the Angle Addition Postulate, alternate
interior angles, and other theorems you know.
© Carnegie Learning
5
5.2 Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems 463
Problem 2
Students use measuring tools
to draw several triangles.
Through answering a series of
questions, students realize the
measure of an interior angle
in a triangle is directly related
to the length of the side of the
triangle opposite that angle. As
an angle increases in measure,
the opposite side increases in
length to accommodate the
angle. And as a side increases
in length, the angle opposite
the side increases in measure
to accommodate the length of
the side.
Grouping
Have students complete
Questions 1 through 6 with a
partner. Then have students
share their responses as
a class.
Guiding Questions for Share Phase, Question 1
What is an acute triangle?
Is the shortest side of your
triangle opposite the angle of
smallest measure?
Do you think this relationship
is the same in all triangles?
Why or why not?
PROBLEM 2 Analyzing Triangles
1. Consider the side lengths and angle measures of an acute triangle.
a. Draw an acute scalene triangle. Measure each interior angle and label the angle
measures in your diagram.
Answers will vary.
70°
3.5 cm3.4 cm
4 cm
56° 54°
b. Measure the length of each side of the triangle. Label the side lengths in
your diagram.
See diagram.
c. Which interior angle is opposite the longest side of the triangle?
The largest interior angle is opposite the longest side of the triangle. In my
diagram, the largest interior angle is 70° and the longest side is 4 centimeters.
d. Which interior angle lies opposite the shortest side of the triangle?
The smallest interior angle is opposite the shortest side of the triangle. In my
diagram, the smallest interior angle is 54° and the shortest side is 3.4 centimeters.
© Carnegie Learning
464 Chapter 5 Properties of Triangles
5
Guiding Questions for Share Phase, Question 2
What is an obtuse triangle?
Is the longest side of your
triangle opposite the angle of
largest measure?
Do you think this relationship
is the same in all triangles?
Why or why not?
2. Consider the side lengths and angle measures of an obtuse triangle.
a. Draw an obtuse scalene triangle. Measure each interior angle and label the angle
measures in your diagram.
Answers will vary.
110°
45°
2.4 cm
5.2 cm
3.9 cm
25°
b. Measure the length of each side of the triangle. Label the side lengths in
your diagram.
See diagram.
c. Which interior angle lies opposite the longest side of the triangle?
The largest interior angle lies opposite the longest side of the triangle. In my
diagram, the largest interior angle is 110° and the longest side is 5.2 centimeters.
d. Which interior angle lies opposite the shortest side of the triangle?
The smallest interior angle lies opposite the shortest side of the triangle. In my
diagram, the smallest interior angle is 25° and the shortest side is 2.4 centimeters.
© Carnegie Learning
5
Guiding Questions for Share Phase, Question 3
What is a right
scalene triangle?
Is the shortest side of your
triangle opposite the angle of
largest measure?
Is the longest side of your
triangle opposite the angle of
largest measure?
Do you think this relationship
is the same in all triangles?
Why or why not?
5.2 Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems 465
3. Consider the side lengths and angle measures of a right triangle.
a. Draw a right scalene triangle. Measure each interior angle and label the angle
measures in your diagram.
Answers will vary.
90°
55° 35°
4 cm
3.2 cm2.3 cm
b. Measure each side length of the triangle. Label the side lengths in your diagram.
See diagram.
c. Which interior angle lies opposite the longest side of the triangle?
The largest interior angle lies opposite the longest side of the triangle. In my
diagram, the largest interior angle is 90° and the longest side is 4 centimeters.
d. Which interior angle lies opposite the shortest side of the triangle?
The smallest interior angle lies opposite the shortest side of the triangle. In my
diagram, the smallest interior angle is 35° and the shortest side is 2.3 centimeters.
© Carnegie Learning
466 Chapter 5 Properties of Triangles
5
Guiding Questions for Share Phase, Questions 4 through 6
How would the side length
of a triangle change if the
measure of its opposite
angle increases?
How would the side length
of a triangle change if the
measure of its opposite
angle decreases?
What is the relationship
between the interior angle
measure of a triangle and its
side lengths?
4. The measures of the three interior angles of a triangle are 57°, 62°, and 61°. Describe
the location of each side with respect to the measures of the opposite interior angles
without drawing or measuring any part of the triangle.
a. longest side of the triangle
The longest side of the triangle is opposite the largest interior angle; therefore,
the longest side of the triangle lies opposite the 62° angle.
b. shortest side of the triangle
The shortest side of the triangle is opposite the smallest interior angle; therefore,
the shortest side of the triangle lies opposite the 57° angle.
5. One angle of a triangle decreases in measure, but the sides of the angle remain the
same length. Describe what happens to the side opposite the angle.
As an interior angle of a triangle decreases in measure, the sides of that angle are
forced to move closer together, creating an opposite side of the triangle that decreases
in length.
6. An angle of a triangle increases in measure, but the sides of the angle remain the same
length. Describe what happens to the side opposite the angle.
As an interior angle of a triangle increases in measure, the sides of that angle are
forced to move farther apart, creating an opposite side of the triangle that increases
in length.
© Carnegie Learning
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5.2 Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems 467
Grouping
Have students complete
Question 7 with a partner.
Then have students share their
responses as a class.
Guiding Questions for Share Phase, Question 7
What is the %rst step when
solving this problem?
Which side is the shortest
side of the triangle? How do
you know?
Which side is the longest
side of the triangle? How do
you know?
How did you determine the
measure of the third interior
angle of the triangle?
Problem 3
Students prove two theorems:
For the two-column proof of
the Exterior Angle Theorem,
students are required to write
both the statements and
the reasons.
The Exterior Angle Inequality
Theorem has two Prove
statements so it must be
done in two separate parts.
In the %rst part of this two-
column proof, the reasons
are provided and students
are required to write only
the statements. This was
done because students may
need some support using
the Inequality Property. In
the second part, students
are expected to write both the statements and reasons. They can use the %rst
part as a model.
7. List the sides from shortest to longest for each diagram.
a.
47°
98°
35°
y
zx
b.
52°
81°
m
n p
47°
x, z, y p, n, m
c.
40°
45°40°
95°112°
d
h ge
f
28°
f, e, g, h, d
PROBLEM 3 Exterior Angles
Use the diagram shown to answer Questions 1 through 12.
1
2
34
1. Name the interior angles of the triangle.
The interior angles are /1, /2, and /3.
2. Name the exterior angles of the triangle.
The exterior angle is /4.
3. What did you need to know to answer Questions 1 and 2?
I needed to know the definitions of interior and exterior angles.
© Carnegie Learning
468 Chapter 5 Properties of Triangles
5
Grouping
Have students complete
Questions 1 through 12 with
a partner. Then have students
share their responses as a class.
Guiding Questions for Share Phase, Questions 1 through 8
Where are interior
angles located?
Where are exterior
angles located?
What is the difference
between an interior angle
and an exterior angle?
How can the Triangle Sum
Theorem be applied to
this situation?
What is the relationship
between angle 3 and
angle 4?
How many exterior
angles can be drawn on a
given triangle?
How are exterior
angles formed?
How many remote interior
angles are associated
with each exterior angle of
a triangle?
4. What does m/1 1 m/2 1 m/3 equal? Explain your reasoning.
m/1 1 m/2 1 m/3 5 180°
The Triangle Sum Theorem states that the sum of the measures of the interior angles
of a triangle is equal to 180°.
5. What does m/3 1 m/4 equal? Explain your reasoning.
m/3 1 m/4 5 180°
A linear pair of angles is formed by /3 and /4. The sum of any linear pair’s angle
measures is equal to 180°.
6. Why does m/1 1 m/2 5 m/4? Explain your reasoning.
If m/1 1 m/2 1 m/3 and m/3 1 m/4 are both equal to 180°, then m/1 1 m/2 1
m/3 5 m/3 1 m/4 by substitution. Subtracting m/3 from both sides of the
equation results in m/1 1 m/2 5 m/4.
7. Consider the sentence “The buried treasure is located on a remote island.” What does
the word remote mean?
The word remote means far away.
8. The exterior angle of a triangle is /4, and /1 and /2 are interior angles of the same
triangle. Why would /1 and /2 be referred to as “remote” interior angles with respect
to the exterior angle?
Considering all three interior angles of the triangle, /1 and /2 are the two interior
angles that are farthest away from, or not adjacent to, /4.
© Carnegie Learning
5
5.2 Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems 469
Guiding Questions for Share Phase, Questions 9 through 12
How many remote interior
angles are associated with
each exterior angle
of a triangle?
What is the difference
between a postulate
and a theorem?
The remote interior angles of a triangle are the two angles that are non-adjacent to the
speci#ed exterior angle.
9. Write a sentence explaining m/4 5 m/1 1 m/2 using the words sum, remote interior
angles of a triangle, and exterior angle of a triangle.
The measure of an exterior angle of a triangle is equal to the sum of the measures of
the two remote interior angles of the triangle.
10. Is the sentence in Question 9 considered a postulate or a theorem? Explain
your reasoning.
It would be considered a theorem because it can be proved using definitions, facts,
or other proven theorems.
11. The diagram was drawn as an obtuse triangle with one exterior angle. If the triangle had
been drawn as an acute triangle, would this have changed the relationship between the
measure of the exterior angle and the sum of the measures of the two remote interior
angles? Explain your reasoning.
No. If /1 and /2 were both acute, the sum of their measures would still be equal
to m/4.
12. If the triangle had been drawn as a right triangle, would this have changed the
relationship between the measure of the exterior angle and the sum of the measures of
the two remote interior angles? Explain your reasoning.
No. If /1 or /2 were a right angle, the sum of their measures would still be equal
to m/4.
© Carnegie Learning
470 Chapter 5 Properties of Triangles
5
Grouping
Have students complete
Question 13 with a partner.
Then have students share their
responses as a class.
Guiding Questions for Share Phase, Question 13
Are any theorems used to
prove this theorem? If so,
which theorems?
Are any de%nitions used to
prove this theorem? If so,
which de%nitions?
Are any properties used to
prove this theorem? If so,
which properties?
The Exterior Angle Theorem states: “the measure of
the exterior angle of a triangle is equal to the sum of the
measures of the two remote interior angles of the
triangle.”
13. Prove the Exterior Angle Theorem using the
diagram shown.
A
B C D
Given: Triangle ABC with exterior /ACD
Prove: m/ A 1 m/B 5 m/ ACD
Statements Reasons
1. Triangle ABC with exterior /ACD 1. Given
2. m/A 1 m/B 1 m/BCA 5 180° 2. Triangle Sum Theorem
3. /BCA and /ACD are a linear pair 3. Definition of linear pair
4. /BCA and /ACD are supplementary 4. Linear Pair Postulate
5. m/BCA 1 m/ACD 5 180° 5. Definition of supplementary angles
6. m/A 1 m/B 1 m/BCA 5
m/BCA 1 m/ACD
6. Substitution Property using step 2
and step 5
7. m/A 1 m/B 5 m/ACD 7. Subtraction Property of Equality
Think about the Triangle Sum
Theorem, the definition of “linear pair,” the Linear Pair Postulate, and other definitions or facts that
you know.
© Carnegie Learning
5
Grouping
Have students complete
Question 14 with a partner.
Then have students discuss
their responses as a class.
Guiding Questions for Share Phase, Question 14
What is the %rst step when
solving for the value of x?
What information in the
diagram helps you determine
additional information?
If you know the measure of
the exterior angle, what else
can be determined?
Is an equation needed to
solve for the value of x? Why
or why not?
What equation was used to
solve for the value of x?
5.2 Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems 471
14. Solve for x in each diagram.
a.
108°
156°
x
b.
x
x
152°
x 1 x 5 152°
x 1 108° 5 156° 2x 5 152°
x 5 48° x 5 76°
c.
120°
3x
2x
d.
(2x + 6°)
126°
x
x 1 2x 1 6° 5 126°
2x 1 3x 5 120° 3x 1 6° 5 126°
5x 5 120° 3x 5 120°
x 5 24° x 5 40°
The Exterior Angle Inequality Theorem states: “the measure of an exterior angle of a
triangle is greater than the measure of either of the remote interior angles of the triangle.”
15. Why is it necessary to prove two different statements to completely prove
this theorem?
I must prove that the exterior angle is greater than each remote interior angle
separately.
© Carnegie Learning
472 Chapter 5 Properties of Triangles
5
Grouping
Have students complete
Questions 15 through 16 part (a)
with a partner. Then have
students share their responses
as a class.
Guiding Questions for Share Phase, Questions 15 through 16 part (a)
Which theorems were used
to prove this theorem?
Which de%nitions were used
to prove this theorem?
Which properties were used
to prove this theorem?
Which angles are the three
interior angles?
Which angle is the
exterior angle?
Which angles are the two
remote interior angles?
Which angles form a linear
pair of angles?
What substitution is
necessary to prove
this theorem?
Is it possible for the measure
of an angle to be equal to 0
degrees? Why or why not?
16. Prove both parts of the Exterior Angle Inequality Theorem using the diagram shown.
A
B C D
a. Part 1
Given: Triangle ABC with exterior /ACD
Prove: m/ACD . m/A
Statements Reasons
1. Triangle ABC with exterior /ACD 1. Given
2. m/A 1 m/B 1 m/BCA 5 180° 2. Triangle Sum Theorem
3. /BCA and /ACD are a linear pair 3. Linear Pair Postulate
4. m/BCA 1 m/ACD 5 180° 4. De#nition of linear pair
5. m/A 1 m/B 1 m/BCA 5
m/BCA 1 m/ACD
5. Substitution Property using
step 2 and step 4
6. m/A 1 m/B 5 m/ACD 6. Subtraction Property of Equality
7. m/B . 0° 7. De#nition of an angle measure
8. m/ACD . m/A 8. Inequality Property (if a 5 b 1 c and
c . 0, then a . b)
© Carnegie Learning
5
Grouping
Have students complete
Question 16, part (b) with a
partner. Then have students
share their responses as a class.
Guiding Questions for Share Phase, Question 16 part (b)
Which theorems were used
to prove this theorem?
Which de%nitions were used
to prove this theorem?
Which properties were used
to prove this theorem?
How is this proof similar to
the proof in part (a)?
How is this proof different
from the proof in part (a)?
5.2 Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems 473
b. Part 2
Given: Triangle ABC with exterior /ACD
Prove: m/ACD . m/B
Statements Reasons
1. Triangle ABC with exterior/ACD 1. Given
2. m/A 1 m/B 1 m/BCA 5 180° 2. Triangle Sum Theorem
3. /BCA and /ACD are a linear pair 3. Linear Pair Postulate
4. m/BCA 1 m/ACD 5 180° 4. Definition of linear pair
5. m/A 1 m/B 1 m/BCA 5
m/BCA 1 m/ACD
5. Substitution Property using
step 2 and step 4
6. m/A 1 m/B 5 m/ACD 6. Subtraction Property of Equality
7. m/A . 0° 7. Definition of an angle measure
8. m/ACD . m/B 8. Inequality Property (if a 5 b 1 c
and c . 0, then a . b)
© Carnegie Learning
474 Chapter 5 Properties of Triangles
5
Problem 4
Students are given two different
maps of Easter Island. Both
maps contain a key in which
distance can be measured
in miles or kilometers. Easter
Island is somewhat triangular
in shape and students predict
and then compute which
side appears to contain the
longest coastline in addition to
answering questions related to
the perimeter of the island.
Grouping
Ask students to read the
information. Discuss as a class.
PROBLEM 4 Easter Island
Easter Island is an island in the southeastern Paci#c Ocean, famous for its statues created
by the early Rapa Nui people.
Two maps of Easter Island are shown.
109° 25' 109° 20' 109° 15'
27° 05'
27° 10'
Vinapu
HangaPoukuia
Vaihu
HangaTe'e
Akahanga
Puoko
Tongariki
Mahutau
Hanga Ho'onuTe Pito Te Kura
Nau Nau
Papa Tekena
Makati Te Moa
Tepeu
Akapu
Orongo
A Kivi
Ature Huku
Huri A UrengaUra-Urangate Mahina
Tu'u-Tahi
Ra'ai
A Tanga
Te Ata Hero
Hanga TetengaRunga Va'e
Oroi
Mataveri
Hanga Piko
Hanga Roa
Aeroportointernazionale
di Mataveri
VulcanoRana Kao
VulcanoRana
Roratka
VulcanoPuakatike
370 mCerro Puhi
302 m
Cerro Terevaka507 m
Cerro Tuutapu270 m
Motu Nui
Capo Sud
Punta Baja
PuntaCuidado
CapoRoggeveen
CapoCumming
CapoO'Higgins
Capo Nord
Punta San Juan
Punta Rosalia
BaiaLa Pérouse
OCEANO PACIFICO
MERIDIONALECaletaAnakena
RadaBenepu
Hutuiti
Punta Kikiri Roa
Punta One Tea
Maunga O Tu'u300 m
194 m
Maunga Orito220 m
VAIHU
POIKE
HATU HI
OROI
Motu Iti
MotuKau Kau
MotuMarotiri
0 3 Km1 2
0 3 Mi1 2
Altitudine in metri
550
500
450
400
350
300
250
200
150
100
50
0
- 25
- 50
- 100
- 200
- 300
strada
pista o sentiero
Ahu (piattaformacerimoniale)
rovine
Vinapu
Isola di Pasqua
(Rapa Nui)
40° S
30° S
20° S
50° O60° O70° O80° O90° O100° O110° O120° O
ARGENTINA
BOLIVIA
URUGUAY
BRASILEPARAG
UAY
SantiagoIsole JuanFernández
Isola Salay Gomez
Isola di Pasqua
San FélixSan
Ambrosio
0 300 km
300 mi0
Make and hand out
copies of a map of a
local park with various
landmarks identi%ed.
Have students work in
pairs. Each student picks
three points on the map
to make a triangle without
letting the other student
know. Each student
asks the other about
the triangle in order to
%gure out which points
they picked. They can
ask questions about
the classi%cation of the
triangle, interior angle
measures, and exterior
angle measures.
© Carnegie Learning
5
5.2 Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems 475
Grouping
Have students complete
Questions 1 though 7 with a
partner. Then have students
share their responses
as a class.
Guiding Questions for Share Phase, Questions 1 through 5
What is the difference
between the two maps?
Are the distances on each
map reasonably close to
each other? How do
you know?
If you had to answer
questions associated with
elevation, which map is
most useful?
How is the map key used?
Do you suppose all maps
contain a map key?
Why or why not?
Which corner of the island
appears to be formed by the
angle greatest in measure?
What does this tell you about
the coastline opposite the
angle of greatest measure?
1. What questions could be answering using each map?
Answers will vary.
If I need to answer questions about the elevation of various locations, I would need
to use the first map.
If I want to compute linear measurements, both maps contain a map key that I can use.
2. What geometric shape does Easter Island most closely resemble? Draw this shape on
one of the maps.
Easter Island most closely resembles a triangle.
3. Is it necessary to draw Easter Island on a coordinate plane to compute the length of its
coastlines? Why or why not?
No. It is not necessary to use a coordinate plane because a map key is provided.
4. Predict which side of Easter Island appears to have the longest coastline and state your
reasoning using a geometric theorem.
The south side of Easter Island lies opposite what appears to be the angle of
greatest measure, so it will have the longest coastline.
5. Use either map to validate your answer to Question 4.
Answers will vary.
© Carnegie Learning
476 Chapter 5 Properties of Triangles
5
Guiding Questions for Share Phase, Questions 6 and 7
What operation is used to
determine the number of
statues per square mile?
What is the approximate
perimeter of Easter Island?
6. Easter Island has 887 statues. How many statues are there on Easter Island per
square mile?
I used the triangle that I drew and the scale to determine that Easter Island is
approximately 69 square miles.
There are 887 4 69, or 12.86, statues per square mile.
7. Suppose we want to place statues along the entire coastline of the island, and the
distance between each statue was 1 mile. Would we need to build additional statues,
and if so, how many?
I used the triangle that I drew and the scale to determine that the coastline of Easter
Island is less than 887 miles, so we would not need to build additional statues.
© Carnegie Learning
5
5.2 Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems 477
Talk the Talk
A diagram is given consisting of
two triangles sharing a common
side. Students determine
the longest line segment in
a diagram using only the
measures of angles provided.
Grouping
Have students complete the
Talk the Talk with a partner.
Then have students share their
responses as a class.
Talk the Talk
Using only the information in the diagram shown, determine which two islands are farthest
apart. Use mathematics to justify your reasoning.
Grape Island
Mango IslandKiwi Island
Lemon Island
90°
32°
58°
32°
43°
105°
Using the Triangle Sum Theorem, the unknown angle measure near Kiwi Island is 105°
and the unknown angle measure near Grape Island is 32°. In the triangle on the right,
the angle near Mango Island is the largest angle in the triangle, so the side opposite this
angle must be the longest side (and is shared by both triangles). In the triangle on the left,
the angle near Kiwi Island is the largest angle in the triangle, so the side opposite this
angle must be the longest side of the triangle. Therefore, the longest side of the two
triangles is the side between Lemon Island and Grape Island.
However, Lemon Island and Mango Island may be the two islands that are farthest apart.
To determine whether Lemon Island and Mango Island are the farthest apart, I would
need to know the angle measures of the triangle formed by Lemon Island, Mango Island,
and Grape Island.
Grape Island
Mango IslandKiwi Island
Lemon Island
90°
32°
58°
32°
43°
105°
Be prepared to share your solutions and methods.
© Carnegie Learning
478 Chapter 5 Properties of Triangles
5
Check for Students’ Understanding
Quadrilateral WXYZ is shown.
W
X
Y
Z
35°
120°25°
30°60°
90°
1. Without using a ruler, determine which line segment is the longest in this %gure? Explain.
In triangle WXZ, /W is the largest angle, so the segment opposite this angle in triangle WXZ
must be the longest side. In triangle XYZ, /Z the largest angle, so the segment opposite this
angle in triangle XYZ must be the longest side. So line segment XY is the longest line segment
in the figure.
Quadrilateral KDRP is shown.
P
R
K
D
4 cm
9 cm
3 cm
3 cm 6 cm
2. Without using a protractor, determine which angle is the largest in triangle KDR. Explain.
/D is the largest angle in triangle KDR because it is opposite the longest side.
3. Without using a protractor, determine which angle is the largest in triangle PRK. Explain.
/R the largest angle in triangle PRK because it is opposite the longest side.
4. Compare the largest angle in triangle KDR to the largest angle in triangle PRK. Which angle is
larger? How do you know?
/PRK is larger than /KDR, because it is opposite the longer side.