Answer Key • Lesson 5: Angles in Polygons

Answer Key ? Lesson 5: Angles in Polygons

Student Guide

Angles in Polygons

Questions 1?16 (SG pp. 396?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?6. Steps should produce an angle sum that is as close to a straight angle as shown:

Angles in Polygons

What is a Polygon? These shapes are polygons.

A A

B

B

C

C

D

F

D

E

F

E

These shapes are not polygons.

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G

G

H H

I I

7?8. Sum of angles should be close to 180?.

Triangle

Angle Measures

Sum of Angles

45? Red Triangle (D) 45? 90?

180?

J

K

J

K

1. What makes a shape a polygon?

60? Blue Triangle (I) 30? Yellow Triangle (J)

60? 60? 30? 120?

180? 180?

396 SG ? Grade 4 ? Unit 9 ? Lesson 5

Student Guide - Page 396

L L

Angles in Polygons

70? 70?

40?

D

Adding the Angles of a Triangle To investigate the angles in a triangle, complete the following questions.

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.

F E

H

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. AngleAmngelaesmureeaissure is writtewnrhiteteren. here.

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

G

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?

8. Compare your results for Questions 6 and 7 with at least two other students' results. How do your results compare?

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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. Measure (or use shortcuts) to find the measures of the angles in the three Power PolygonsTM below. Fill in a table like the one below to find the sums of the angles of those triangles.

D

red triangle

J

yellow triangle

I

blue triangle

Triangle Red Triangle ( D ) Blue Triangle ( I ) Yellow Triangle ( J )

Angle Measures

Sum of Angles

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

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?.

11. Why do you think the three students' measurements were different from each other?

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.

Angles in Polygons

SG ? Grade 4 ? Unit 9 ? Lesson 5 399

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Student Guide - Page 399

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

The Talking Triangles by: Professor Peabody

I am triangle DEF.

I am triangle ABC.

I have a right angle,

I have only acute

so I am a right

angles, so I am an

triangle. My right

TachuetTeThaterlkiTaiannglgkleiT.nrgiaTnrgialensgalensgle is E.

by: bPyr:ofPersosfoeBrssPoeraPbeoadbyody

I am triangle GHI. I have an obtuse angle, so I am an obtuse triangle.

I amI tarmiantgrlieanAgBleC.ABC.

I haIvehaovnelyoanclyutaecute

angleasn,gsleos,I saomI aanm Aacutaecuttreiantgrilaen. gle.

aCn

I amI tarmiantgrlieanDgEleF.DEF.

I haIvehaavreigahtrigahntglea,ngle, so I saomI aamrigahtright triantgrilaen. gMley. rMigyhtright angleanigsleiEs. E.

I amI tarmiantgrlieanGgHleI.GHI. I haIvehaavneoabntuosbetuse anglea,ngsole,I saomI aanm an obtuosbetutsreiantgrilaen. gle.

13. A. Do you agreBe thBat triangle ABC is an acute triangle? How do you know?

BE. Use a ruler to mFake a sketch of theGtwo missing triangles on a separHate sheet of pa5p3?er. Label the vertices with t3h7e? correct letters. 35?

14.

Can A

aAn

obtuse

triangleC

haCve

a

right

angle?

Explain.

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

the angles.

I

AE. E

FF

53? 53?

B. G G

37? 37?

HH

35? 35?

D

II

DD

9. Angle measures and sums should be close to the following:

Triangle Red Triangle (D)

Angle Measures

45? 45? 90?

60? Blue Triangle (I) 60? 60?

30? Yellow Triangle (J) 30? 120?

Sum of Angles

180? 180? 180?

Triangle

Angle Measures

Sum of Angles

10R. edAT?riaBng.le (DA) nsw4e5r?s wil4l 5v?ary, b9u0t?should i1n8c0lu?de the re70su? lt that the sum of the angles in any

Blue Triantrgilea(nI)gle 6is01? 80?.60? 60?

180?

70?

11Y.e*lloEw rT4rr0iao?nrgsle a(Jr) e l3ik0e?ly in3m0?easu1re2m0?ent. 180?

12. A. Yes.

D

70?

Triangle

Angle Measures

Sum of Angles

E

Triangle Red Triangle

(D7) 0?

R4ed0?Triangle (D) F

Angle Measures

45? 45? 90? 45? 45? 90?

Sum of Angles

180? 180?

60? Blue Triangle (I) 60? 60?

180?

B.BluNe oTri,antgwleo(I)righ6t0a?ngles60h?ave a6s0u?m of 18108?0?

D

H

YelalolwreTraiadngyle.

Yellow Triangle

((JTJ)) he33t0r0i??angle3300c??anno11t22b00e??comple11t88e00d??

with a third angle.

G C.* No. Similar reasIoning to Part B.

13. EA. Yes. Expla77Fn00??ations will vary. Possible explanation: All the angles are less

70?

than409?0H?. 70? B. Poss4i0b?le right triangle:

GD

I

D

F

E

F

E

Possible obtuse triangle:

H

H

G

I

G

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?

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400 SG ? Grade 4 ? Unit 9 ? Lesson 5

Angles in Polygons

Student Guide - Page 400

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

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

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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: --A--E makes a straight angle. I know AFC = 30?, so 180? ? 30? = 150?.

A

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

A. What is the measure of A? B. What is the measure of AGB? C. What is the measure of AFC? D. What is the measure of EFC? E. Look at polygon EFCD. What is the sum of all the angles?

Use the Add and Subtract Angles pages in the Student Activity Book to practice finding unknown angle measurements.

B

C

D

A

F 30?

C

150?

E

E. Polygon EFCD = 180?. Possible explanation: 90? + 90? + 30? + 150? = 360?.

Student Guide

Homework

Questions 1?8 (SG pp. 401?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.

E F

H G

1. Which of these shapes are polygons?

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

Angles in Polygons

SG ? Grade 4 ? Unit 9 ? Lesson 5 401

Student Guide - Page 401

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

A.

D DDD

167? 111666777??? ?

???

D

A AAA B BBB 167C? CCC ?

A

B

C

B.

E EEE

?

??E?

36? 333666??? ?

F FFF G 36GG?G

F

G

CS. SSS T TTT U UUU

? S 11??1?? 111111111??? T

U

V

VVV

? 111?

V

D. W WWW W? ???

? 49? 44949?9??

X XXX Y 4YY9Y? Z ZZZ

X

Y

Z

4. Is hexaQgon LQQMQNOPQ LregulaLLrL? Why or why not?

Q

L

P PPP P

M MMM M

O OOO

N NNN

O

N

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402 SG ? Grade 4 ? Unit 9 ? Lesson 5

Angles in Polygons

Student Guide - Page 402

TG ? Grade 4 ? Unit 9 ? Lesson 5 ? Answer Key 3

Answer Key ? Lesson 5: Angles in Polygons

B

A

E

C

B BB

A AA

D

E EE

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 i3n5?the picture above.

27?

68?

C. Find a1n15o?bt4u2s?e 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.

6.

Find the angles.

missing

angle

measures

in

theDtrDiDangles

below.

Do

not

measure

the

A.

B.

C.

42? 1151?11155??

4422??

What do you call a lost parrot?

35? 3355??

27? 2277??

68? 6688??

A "polygon."

Angles in Polygons

SG ? Grade 4 ? Unit 9 ? Lesson 5 403

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?

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Student Guide - Page 403

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

180?

180?

D D

E E

20?

A

B

C

20?

A

B

C

S

T

8. Find the measurements6fo0r? all the angles

S

T

in the figure. Do not use your protractor.

60?

A. T

B. V

C. TSV

D. TUV

E. VSU

F. SUT

G. SUV

V

U

V

U

404 SG ? Grade 4 ? Unit 9 ? Lesson 5

Student Guide - Page 404

Angles in Polygons

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

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