Answer Key • Lesson 5: Angles in Polygons

嚜澤nswer Key ? Lesson 5: Angles in Polygons

Student Guide

Angles in Polygons

Angles in Polygons

Questions 1每16 (SG pp. 396每401)

1.* Answers will vary. Students may reply that

polygons are made of line segments connected

at endpoints, and that each endpoint meets the

endpoint of only one other side with no

overlap.

2每6. Steps should produce an angle sum that is as

close to a straight angle as shown:

What is a Polygon?

These shapes are polygons.

B

A

C

F

D

E

D

These shapes are not polygons.

H

G

7每8. Sum of angles should be close to 180∼.

Angle Measures

Red Triangle (D)

45∼

45∼

90∼

180∼

Blue Triangle (I)

60∼

60∼

60∼

180∼

Yellow Triangle (J)

30∼

30∼

120∼

180∼

I

H

J

I

L

K

J

Sum of Angles

F

E

K

Copyright ? Kendall Hunt Publishing Company

G

Triangle

C

B

A

L

1. What makes a shape a polygon?

396

SG ? Grade 4 ? Unit 9 ? Lesson 5

Angles in Polygons

Student Guide - Page 396

70∼

Adding the Angles of a Triangle

70∼

To investigate the angles in a triangle, complete the following questions.

40∼

2. Use a pencil and ruler to draw a triangle on a sheet of blank paper. Make

sure the triangle covers at least one fourth of the paper. Mark the vertices

with dots as shown in the example.

D

3. Measure the three angles of the triangle with your protractor. Write the angle

measures inside the triangle. Cut out the triangle.

Angle measure is

written here.

measure

is

AngleAngle

measure

is

written

written

here. here.

F

H

G

4. Tear off the three angles as shown in the example.

I

5. Place the three angles next to each other with the vertices and edges

touching each other as shown.

6. Describe the angle that is formed when all three angles are placed together

this way.

7. Add the angle measures of each angle in your triangle. What is the sum of

the three angles?

Copyright ? Kendall Hunt Publishing Company

Copyright ? Kendall Hunt Publishing Company

E

8. Compare your results for Questions 6 and 7 with at least two other

students＊ results. How do your results compare?

398

SG ? Grade 4 ? Unit 9 ? Lesson 5

Angles in Polygons

Student Guide - Page 398

*Answers and/or discussion are included in the lesson.

TG ? Grade 4 ? Unit 9 ? Lesson 5 ? Answer Key

1

Answer Key ? Lesson 5: Angles in Polygons

9. Angle measures and sums should be close to the

following:

9. Measure (or use shortcuts) to find the measures of the angles in the three

Power Polygons? below. Fill in a table like the one below to find the sums

of the angles of those triangles.

Triangle

J

Angle Measures

45∼

45∼

90∼

180∼

Blue Triangle (I)

60∼

60∼

60∼

180∼

Yellow Triangle (J)

30∼

30∼

120∼

180∼

D

yellow triangle

Sum of Angles

Red Triangle (D)

I

red triangle

blue triangle

Triangle

Triangle

Angle Measures

Sum of Angles

Red Triangle ( D )

Blue Triangle ( I )

Yellow Triangle ( J )

10. A. Write a rule about the sum of the angles in a triangle.

Copyright ? Kendall Hunt Publishing Company

B. Do you think the rule is true for every triangle? Explain your thinking.

The students in Mrs. Dewey＊s class answered Question 7 using their own triangles.

Nila found the sum of the angles in her triangle to be 180∼. When Romesh added

the angles in his triangle, the sum was 179∼. Ana＊s sum was 182∼.

70∼

12. Think about the kinds of angles that make up the 180 degrees in a triangle.

A. Can a triangle have all acute angles? If so, show an example.

B. Can a triangle have all right angles? More than one right angle? Show or

tell how you know.

C. Can a triangle have more than one obtuse angle? Show or tell how you

know.

SG ? Grade 4 ? Unit 9 ? Lesson 5

399

Student Guide - Page 399

Professor Peabody forgot to finish drawing the triangles in his cartoon below.

The

Triangles

T

he Talking

Ta

Talking Tr

T

riangles

by:

Peabody

b

y: Professor

Professor P

eabo dy

Talking

Triangles

TheThe

Talking

Triangles

Professor

Peabody

by: by:

Professor

B Peabody

am triangle

I amI triangle

ABC.ABC.

I have

acute

I have

only only

acute

angles,

am an

angles,

so I so

amI an

C

Aacute

acute

triangle.

triangle.

B

Triangle

Triangle

180∼

90∼

45∼

45∼

180∼

90∼

45∼

45∼

Blue Triangle (I)

60∼

60∼

60∼

180∼

Blue

Triangle

(I)right

60∼

60∼

60∼

180∼

B.

No,

two

angles

have

a

sum

of

180∼

H

D

120∼

180∼

30∼

30∼

Yellow

Triangle (J)

already.

The30∼

triangle30∼

cannot

be completed

120∼

180∼

Yellow Triangle (J)

with a third angle.

G

I

C.* No. Similar reasoning

to Part B.

F

70∼

13. EA. Yes. Explanations

will vary.

70∼

Possible explanation: All the angles are less

70∼

H

than40∼

90∼.

70∼

40∼

B. Possible right triangle:

G

am triangle

I amI triangle

DEF. DEF.

I have

a right

angle,

I have

a right

angle,

am triangle

I amI triangle

GHI.GHI.

a right

so I so

amI aam

right

I have

an obtuse

an obtuse

triangle.

My right I have

triangle.

My right

angle,

am an

angle,

so I so

amI an

angleangle

is is

E. E.

obtuse

triangle.

obtuse

triangle.

H

H

F

H

B.E Use a ruler to make

a sketch of theGtwo missing triangles on a separate

37? correct letters.

35?

53?

sheet of paper.

Label the vertices with the

F

F

53? 53?

B.

G

G

H

37? 37?

H

35? 35?

D

I

D

400

I

Copyright ? Kendall Hunt Publishing Company

A

E

D

SG ? Grade 4 ? Unit 9 ? Lesson 5

G

G

I

I

14. No. The sum of the obtuse angle and a right

angle is already greater than the sum of all

three angles in a triangle, 180∼.

15. A.* ＋D = 37∼

B. ＋I = 108∼

Angles in Polygons

Student Guide - Page 400

*Answers and/or discussion are included in the lesson.

2

F

F

Possible obtuse triangle:

B that triangle ABC is an acute triangle? How do you know?

13. A. Do you agree

A.E

I

D

D

E

E

15. Find the missing degree measures for the triangles below. Do not measure

the angles.

I

Sum of Angles

Sum of Angles

E

triangle

GHI..

I am tr

iangle GHI

I ha

ve an obtu

se

have

obtuse

angle,

an

gle, so I am an

obt

use ttriangle.

riangle.

obtuse

14. Can an

C a right angle? Explain.

C have

A obtuse triangle

Angle Measures

Angle Measures

Red Triangle (D)70∼

Red

40∼Triangle (D) F

TG ? Grade 4 ? Unit 9 ? Lesson 5 ? Answer Key

Copyright ? Kendall Hunt Publishing Company

triangle

I am tr

iangle DEF.

DEF.

I ha

ve a rright

ight an

gle,

have

angle,

so I am a rright

ight

tr

iangle. My rright

ight

triangle.

an

gle is E.

angle

triangle

ABC..

I am tr

iangle ABC

have

acute

I ha

ve oonly

nly a

cute

an

gles, so I am an

angles,

a

cute tr

iangle.

acute

triangle.

Sum of Angles

D

11. Why do you think the three students＊ measurements were different from

each other?

Angles in Polygons

Angle Measures

Triangle (D)

10.RedA每B.

Answers

vary, but

180∼

90∼should include

45∼ will45∼

70∼

the result that the sum of the angles in any

Blue Triangle (I)

60∼

180∼

triangle 60∼

is 180∼.60∼

70∼

40∼ are

120∼

180∼

30∼ in30∼

11.*

Errors

likely

measurement.

Yellow

Triangle

(J)

12. A. Yes.

Answer Key ? Lesson 5: Angles in Polygons

16. A. ＋A = 60; Possible explanation:

＋D + ＋E + ＋A = 180∼

so 90∼ + 30∼ + 60∼ = 180∼

B. ＋AGB = 30∼; Possible explanation:

＋A + ＋G + ＋B = 180∼;

60∼ + ＋G + 90∼ = 180∼;

so ＋AGB = 30∼.

C. ＋AFC = 30∼; Possible explanation:

＋A = 60∼, ＋C = 90∼, so ＋F = 30∼,

D. ＋EFC = 150∼; Possible explanation:

〞

〞

AE makes a straight angle. I know

＋AFC = 30∼, so 180∼ 每 30∼ = 150∼.

16. Find the missing degree measures in the triangles below. Do not measure

the angles. Show or tell how you calculated the measure of each angle.

A

G

B

F

C

30?

E

D

What is the measure of ＋A?

What is the measure of ＋AGB?

What is the measure of ＋AFC?

What is the measure of ＋EFC?

Look at polygon EFCD. What is the sum of all the angles?

A.

B.

C.

D.

E.

Use the Add and Subtract Angles pages in the Student Activity Book to practice

finding unknown angle measurements.

A

30∼

Copyright ? Kendall Hunt Publishing Company

F

B

C

150∼

C

D

A

H

E

F

G

1. Which of these shapes are polygons?

E

2. Which are not polygons? Explain why each is not a polygon.

Angles in Polygons

E. Polygon EFCD = 180∼. Possible

explanation: 90∼ + 90∼ + 30∼ + 150∼ = 360∼.

SG ? Grade 4 ? Unit 9 ? Lesson 5

401

Student Guide - Page 401

Student Guide

3. Without a protractor, find the measure of each missing angle. Name the

missing angle with your measurement.

Homework

D

D

D

D

167?

167?

167? ?

167?

A

AA

A

B

BB

B

D

??

?

C

C

C?

167?

C

A

C

B

B.

E

EE

E

??

E?

?

?

36?

36?

36?

36?

F

S

SS

C.

S

?

V

F

FF

G

G

36?

GG

T

F

T

TT

U

S 111?

?

111?

111?

111?

??

U

T

111?

?

V

VV

G

U

U

U

V

D.

W

W

W

W

??

?

W

?

49?

Copyright ? Kendall Hunt Publishing Company

Copyright ? Kendall Hunt Publishing Company

A.

Questions 1每8 (SG pp. 401每404)

1. B, D, F, and H are polygons.

2. A has a curved side.

C has two segments with unconnected ends.

E has two segments that touch other segments

at three points.

G has curved sides.

3. A. 13∼

B. 54∼

C. 69∼

D. 131∼

4. Hexagon LMNOPQ is not regular. The angles

are not all equal. ＋P and ＋M are acute.

＋L, ＋N, ＋O, and ＋Q are obtuse.

?

49?

49?

49?

Z

ZZ

Z

X

Y

4. Is hexagon LMNOPQ

regular?

Why or why not?

Q

L

QQ

LL

Q

L

Q

L

X

XX

X

P

Y

49?

Y

Y

Z

Y

P

PP

M

P

M

O

O

OO

N

N

NN

O

402

M

M

M

SG ? Grade 4 ? Unit 9 ? Lesson 5

N

Angles in Polygons

Student Guide - Page 402

TG ? Grade 4 ? Unit 9 ? Lesson 5 ? Answer Key

3

Answer Key ? Lesson 5: Angles in Polygons

5. A. Right triangles: triangles BCA and BCE

B. Acute triangles: triangle AED, BAD, and

ACD

C. Obtuse triangles: triangles AEB, DEC, and

BCD

D. Additional examples will vary.

6. A. 23∼

B. 55∼

C. 85∼

7. 70∼

8. A. 90∼

B. 90∼

C. 90∼

D. 90∼

E. 30∼

F. 30∼

G. 60∼

B

E

A

C

B BB

D

E EE

A AA

C CC

5. A. Find a right triangle in the picture above. Name your triangle using the

letters of the triangle＊s three vertex points.

B. Find an acute triangle in

35?the picture above.

27?

68?

42?

115?

C. Find an

obtuse triangle in the picture above.

D. Can you find more examples of right, acute, and obtuse angles in the

picture? Name as many of each as you can.

Copyright ? Kendall Hunt Publishing Company

6. Find the missing angle measures in theDtriangles

below. Do not measure the

DD

angles.

A.

B.

C.

27? 27?

27?

35? 35?

35?

68? 68?

68?

42? 42?

42?

115?115?

115?

What do you call

a lost parrot?

A ※polygon.§

Angles in Polygons

SG ? Grade 4 ? Unit 9 ? Lesson 5

403

Student Guide - Page 403

7. Without using a protractor, find the measure of ＋DBA.

180?

180?

D

E

E

20?

A

V

T

S

SG ? Grade 4 ? Unit 9 ? Lesson 5

T

60?

U

V

404

C

B

Copyright ? Kendall Hunt Publishing Company

S

8. Find the measurements 60?

for all the angles

in the figure. Do not use your protractor.

A. ＋T

B. ＋V

C. ＋TSV

D. ＋TUV

E. ＋VSU

F. ＋SUT

G. ＋SUV

20?

C

B

A

U

Angles in Polygons

Student Guide - Page 404

4

TG ? Grade 4 ? Unit 9 ? Lesson 5 ? Answer Key

Copyright ? Kendall Hunt Publishing Company

D

Answer Key ? Lesson 5: Angles in Polygons

Student Activity Book

Name

Date

Measuring Polygon Angles

Measuring Polygon Angles

Questions 1每10 (SAB pp. 329每331)

1.*

60∼

Here are the names for ten Power Polygons?. Take out these polygons to

help you answer Questions 1每3 below.

60∼

O

N

M

K

brown rhombus

N

small

green triangle

blue rhombus

red trapezoid

60∼

C

J

2.*

120∼

L

60∼

yellow triangle

blue rectangle

orange triangle

M

H

F

3.

120∼

90∼

90∼

90∼

90∼

B

90∼

C

90∼

90∼

90∼

Copyright ? Kendall Hunt Publishing Company

60∼

B

brown triangle

yellow hexagon

orange square

1. Find the measures of the angles of the small green triangle. Write the angle

measures inside the green triangle above.

2. Place two green triangles on top of the blue rhombus. What is the degree

measure of each of the angles of the rhombus? (Hint: you do not have to

measure.) Write the angle measuresHinside the blue rhombus above.

3. Find the angle measures of all the polygons. You can use shortcuts to find

angle measures. Write the angle measures inside the pictured shapes.

yellow hexagon

90∼

45∼

120∼

120∼

Angles in Polygons

SAB ? Grade 4 ? Unit 9 ? Lesson 5

329

F

K

30∼

Student Activity Book - Page 329

red trapezoid

45∼

120∼

120∼

H

O

120?  ? = 180?

O

60∼

M

O

120?

120∼

120∼

120∼

120∼

K

J

60∼

90∼

L

Copyright ? Kendall Hunt Publishing Company

30∼

150∼

60∼

O

150∼

30∼

120∼

30∼

30∼

*Answers and/or discussion are included in the lesson.

TG ? Grade 4 ? Unit 9 ? Lesson 5 ? Answer Key

5

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