Properties of Triangles 5 - IHS Math- Satre - Homesatreihs.weebly.com/uploads/5/8/3/6/58366617/teachers...13 – 18 Identify interior, exterior, and remote interior angles of triangles

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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


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






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.


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



5


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


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.



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.


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.


461D Chapter 5 Properties of Triangles

5


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


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



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





a class.


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?


between angle 1 and angle 4?


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.


5


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


Questions 1 through 6 with a



a class.


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.



5


What is an obtuse triangle?

Is the longest side of your


largest measure?



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


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.


The smallest interior angle lies opposite the shortest side of the triangle. In my



5


What is a right

scalene triangle?

Is the shortest side of your


largest measure?

Is the longest side of your


largest measure?



Why or why not?


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


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.


The smallest interior angle lies opposite the shortest side of the triangle. In my




5


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?


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.


5


Grouping






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.



5

Grouping


Questions 1 through 12 with

a partner. Then have students



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?


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.


5



How many remote interior

angles are associated with

each exterior angle

of a triangle?


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.



5

Grouping






Are any theorems used to

prove this theorem? If so,

which theorems?

Are any de%nitions used to


which de%nitions?

Are any properties used to


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.


5

Grouping



Then have students discuss

their responses as a class.


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?


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.



5

Grouping


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


Which properties were used


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


Prove: m/ACD . m/A

Statements Reasons

1. Triangle ABC with exterior /ACD 1. Given


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


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)


5

Grouping


Question 16, part (b) with a



Guiding Questions for Share Phase, Question 16 part (b)

Which theorems were used


Which de%nitions were used


Which properties were used


How is this proof similar to

the proof in part (a)?

How is this proof different

from the proof in part (a)?


b. Part 2


Prove: m/ACD . m/B

Statements Reasons

1. Triangle ABC with exterior/ACD 1. Given


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


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)



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.


5


Grouping


Questions 1 though 7 with a


share their responses

as a class.



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.



5


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.


5


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.



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.



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.

Documents

Properties of Triangles 5 - IHS Math- Satre - Homesatreihs.weebly.com/uploads/5/8/3/6/58366617/teachers...13 – 18 Identify interior, exterior, and remote interior angles of triangles