Properties of Triangles 5

© 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.


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


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


452 Chapter 5 Properties of Triangles

5


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.


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.


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.


453D Chapter 5 Properties of Triangles

5


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.



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.



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).



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


Questions 1 and 2 with a


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?



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.



5

Grouping


Questions 3 and 4 with a


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.



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.



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


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.

Properties of Triangles 5

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