skills practice for lesson 4 - ms. hollingsworth - homech...exterior angle of a triangle is greater...

Chapter 4 ● Skills Practice 429

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Skills Practice Skills Practice for Lesson 4.1

Name _____________________________________________ Date ____________________

Interior and Exterior Angles of a Triangle Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems

VocabularyWrite the term that best completes each statement.

1. The states that the measure of an

exterior angle of a triangle is greater than the measure of either of the remote

interior angles of the triangle.

2. The states that the sum of the

measures of the interior angles of a triangle is 180°.

3. The states that the measure of an

exterior angle of a triangle is equal to the sum of the measures of the remote

interior angles of the triangle.

4. The are the two angles that are

non-adjacent to the specified exterior angle.

Problem SetDetermine the measure of the missing angle in each triangle.

1.

A

B

C78° 37°

2. P80˚ 66˚

Q

R

m�B � 180° � (78° � 37°) � 65°

430 Chapter 4 ● Skills Practice

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3.

35˚

28˚

K

LM

4.

90˚ 32˚

F E

G

5.

60˚

60˚

W

X

Y

6.

110˚

35˚

T

V U

List the side lengths from shortest to longest for each diagram.

7.a

b

c

48°

21°

A

B

C 8.

60˚ 54˚

S

T

r t

s R

m�C � 180° � (48° � 21°) � 111°

The shortest side of a triangle is opposite the smallest angle. So, the side lengths from shortest to longest are a, b, c.


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9.

28˚118˚

M

K

L

k

l

m

10.

42˚

84˚

Z

YX

yx

z

11.

64˚

79˚67˚

27˚

X Y

W Ze

dca

b 12.

50˚

30˚

60˚90˚

A

u

v

s

r

D

CB

t


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Identify the interior angles, the exterior angle, and the remote interior angles of each triangle.

13.

WX

Y

Z

14.

R S

UT

Interior angles: �XYZ, �YZX, �ZXY

Exterior angle: �WXZ

Remote interior angles: �XYZ, �YZX

15.

E

F

G H

16. B

C

A

D

17. L

MKJ

18.

Q

P

SR


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Solve for x in each diagram.

19.

F

Gx

H

J

K

130°

99°

20.V

S

TU

R 140˚

132˚

x

m�GFH � 180° � 130° � 50°

m�GHK � m�GFH � m�FGH 99° � 50° � x 49° � x

21.

H

I

JK

81˚

x2x

22.

R T V S

U

64˚

90˚ (x + 8)


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23.

132˚

112˚

(2x + 4)

K

L

J

N

M

24.

90˚

(3x + 2) (2x + 18)

F

G

D E

Use the given information for each triangle to write two inequalities that you would need to prove the Exterior Angle Inequality Theorem.

25.

S

RQ

T 26.

S

Q

R

P

Given: Triangle RST with exterior �TRQ Given: Triangle QRS with exterior �PQR

Prove: m�TRQ � m�S and Prove:

m�TRQ � m�T


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27. T

U

VW

28. J

F G H

Given: Triangle UVW with exterior �TUV Given: Triangle GHJ with exterior �FGJ

Prove: Prove:

29. K L M

N

30.

D

C

A B

Given: Triangle LMN with exterior �KLN Given: Triangle ABC with exterior �BCD

Prove: Prove:


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Name _____________________________________________ Date ____________________

Installing a Satellite Dish Simplifying Radicals, Pythagorean Theorem, and Its Converse

VocabularyMatch each definition to its corresponding term.

1. an expression that involves a radical sign a. square root

2. the symbol √__

b. radical sign

3. a number b such that b2 � a c. radicand

4. the sides of a right triangle that form the right angle d. radical expression

5. the expression written under a radical sign e. simplest form

6. when the radicand of a radical expression f. hypotenuse

contains no factors that are perfect squares

7. the side opposite the right angle in a right triangle g. legs


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Problem SetCalculate the value of each radical expression.

1. √___

81 2. √____

121

81 � 92

√___

81 � 9

3. √___

16 √___

16 4. √__

9 √__

9

5. √___

16 √___

36 6. √___

25 √____

100

Simplify each expression and write the result in radical form.

7. √___

63 8. √___

72

√___

63 � √______

9 � 7 � √__

9 √__

7 � 3 √__

7

9. √___

75 10. √___

45

11. √____

300 12. √____

275

Name the form of 1 that you would use to simplify each fraction.

13. 1 ___ √

__ 5 14. 7 ___

√__

7

√

__ 5 ___

√__

5

15. 10 ___ √

__

3 16. 4 ____

√___

19


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Simplify each fraction.

17. 5 ___ √

__

6

5 ___ √

__ 6 � 5 ___

√__

6 �

√__

6 ___ √

__ 6 � 5 √

__ 6 _____

√__

6 √__

6 � 5 √

__ 6 ____

√___

36 � 5 √

__ 6 ____

6

18. 3 ___ √

__ 5

19. 2 ____ √

___

10

20. 4 ___ √

__

6

21. 3 ___ √

__

9

22. 6 ___ √

__

2

Given the area A of a square, calculate the length of one side.

23. A � 48 cm2

A � � 2

� � √__

A � � √

___ 48 � √

_______ 16 � 3 � √

___ 16 √

__ 3 � 4 √

__ 3

Each side is 4 √__

3 centimeters long.

440 Chapter 4 l Skills Practice

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24. A � 75 m2

25. Ingrid covers the floor of a square room with 196 large tiles. The area of each tile

is 1 square foot. What is the length of one side of the room?

26. Devon prepares a square garden with an area of 180 square feet. How much

fencing will Devon need for each side of the garden?

Determine the length of the hypotenuse of each triangle. Round your answer to the nearest tenth, if necessary.

27. 3

c

4

28. 6

c

8

c2 � 32 � 42

c2 � 9 � 16 c2 � 25 c � √

___ 25 � 5

The length of the hypotenuse is5 units.

Chapter 4 l Skills Practice 441

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29.

8 c

8

30. 10

c

15

Determine the length of the unknown leg. Round your answer to the nearest tenth, if necessary.

31. 5

13

b

32. a

15

12

52 � b2 � 132

25 � b2 � 169 b2 � 169 � 25 b2 � 144 b � √

____ 144

b � 12

The length of the unknown leg is 12 units.


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33.

a 10

5

34. 3

9

b

Use the converse of the Pythagorean Theorem to determine whether each triangle is a right triangle. Explain your answer.

35. 8

17

15

Yes. This is a right triangle.

82 � 152 � 64 � 225 � 289 � 172

The sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse, so this is a right triangle.


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36.

59

7

37. 4 10

8

38. 30

50

40


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Use the Pythagorean Theorem to calculate each unknown length. Round your answer to the nearest tenth, if necessary.

39. Chandra has a ladder that is 20 feet long. If the top of the ladder reaches 16 feet

up the side of a building, how far from the building is the base of the ladder?

162 � b2 � 202

256 � b2 � 400 b2 � 400 � 256 b2 � 144 b � √

____ 144

b � 12

The base of the ladder is 12 feet from the building.

40. A scaffold has a diagonal support beam to strengthen it. If the scaffold is 12 feet

high and 5 feet wide, how long must the support beam be?

41. The length of the hypotenuse of a right triangle is 40 centimeters. The legs of the

triangle are the same length. How long is each leg of the triangle?


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42. A carpenter props a ladder against the wall of a building. The base of the ladder is

10 feet from the wall. The top of the ladder is 24 feet from the ground. How long is

the ladder?


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Name _____________________________________________ Date ____________________

Special Right Triangles Properties of a 45° – 45° – 90° Triangle

VocabularyDefine each term in your own words.

1. 45° – 45°– 90° triangle

2. 45° – 45° – 90° Triangle Theorem

Problem SetDetermine the length of the hypotenuse of each 45°– 45°– 90° triangle. Write your answer as a radical in simplest form.

1. 2 in. c

2 in.

2. 5 cm c

5 cm

c � 2 √__

2

The length of the hypotenuse is2 √

__ 2 inches.

3. 9 ft c

9 ft

4. 7 km c

7 km


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Determine the lengths of the legs of each 45°–45°–90° triangle. Write your answer as a radical in simplest form.

5 . a 16 cm

a

6. a 12 mi

a

a √__

2 � 16

a � 16 ___ √

__ 2

a � 16 √__

2 _____ √

__ 2 √

__ 2

a � 16 √__

2 _____ 2 � 8 √

__ 2

The length of each leg is 8 √__

2 centimeters.

7. a 6 2 ft

a

8. a

8 2 m

a

Use the given information to answer each question. Round your answer to the nearest tenth, if necessary.

9. Soren is flying a kite on the beach. The string forms a 45º angle with the ground.

If he has let out 16 meters of line, how high above the ground is the kite?

a √__

2 � 16

a � 16 ___ √

__ 2

a � 16 √__

2 _____ √

__ 2 √

__ 2

a � 16 √__

2 _____ 2 � 8 √

__ 2 � 11.3

The kite is approximately 11.3 meters above the ground.


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10. Meena is picking oranges from the tree in her yard. She rests a 12-foot ladder

against the tree at a 45º angle. How far is the top of the ladder from the ground?

11. Emily is building a square bookshelf. She wants to add a diagonal support beam to

the back to strengthen it. The diagonal divides the bookshelf into two 45º– 45º– 90º

triangles. If each side of the bookshelf is 4 feet long, what must the length of the

support beam be?

12. Prospect Park is a square with side lengths of 512 meters. One of the paths

through the park runs diagonally from the northeast corner to the southwest corner,

and divides the park into two 45º– 45º– 90º triangles. How long is that path?


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Determine the area of each triangle.

13. a 16 mm

a

a √__

2 � 16

a � 16 ___ √

__ 2

a � 16 √__

2 _____ √

__ 2 √

__ 2

a � 16 √__

2 _____ 2

a � 8 √__

2

A � 1 __ 2 (8 √

__ 2 )(8 √

__ 2 )

A � 64( √

__ 2 )2

_______ 2

A � 64(2)

_____ 2

A � 64

The area of the triangle is 64 square millimeters.

14. a 18 in.

a

15. a 7 ft

a


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16. a 11 m

a

Use the given information to answer each question.

17. Eli is making a mosaic using tiles shaped like 45º– 45º– 90º triangles. The length

of the hypotenuse of each tile is 13 centimeters. What is the area of each tile?

a √__

2 � 13

a � 13 ___ √

__ 2 �

13( √__

2 ) _______

√__

2 ( √__

2 )

a � 13 √__

2 _____ 2

A � 1 __ 2 ( 13 √

__ 2 _____

2 ) ( 13 √

__ 2 _____

2 )

A � 169( √

__ 2 )2

________ 8 �

169(2) ______

8

A � 169 ____ 4 � 42.25

The area of each tile is 42.25 square centimeters.

18. Baked pita chips are often in the shape of 45º– 45º– 90º triangles. Caitlyn finds that

the longest side of a pita chip in one bag measures 3 centimeters. What is the area

of the pita chip?


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19. Annika is making a kite in the shape of a 45º– 45º– 90º triangle. The longest side of

the kite is 28 inches. What is the area of the piece of fabric needed for the kite?

20. A tent has a mesh door that is shaped like a 45º– 45º– 90º triangle. The longest side

of the door is 36 inches. What is the area of the mesh door?


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Construct each isosceles triangle described using the given segment.

21. Construct right isosceles triangle ABC with segment BC as the hypotenuse by

constructing 45° angles at B and C.

B C

B

A

C

22. Construct right isosceles triangle WXY with segment WX as the hypotenuse by

constructing 45° angles at W and X.

W X


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23. Construct right isosceles triangle PQR with ___

RQ as a leg and �R as the right angle.

R Q

24. Construct right isosceles triangle DEF with ___

DF as a leg and �D as the right angle.

D F


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Name _____________________________________________ Date ____________________

Other Special Right Triangles Properties of a 30° – 60° – 90° Triangle

VocabularyWrite the term that best completes each statement.

1. A(n) triangle is formed by dividing an equilateral

triangle in half by its altitude.

2. The states that the length of the hypotenuse in

a 30°- 60°- 90° triangle is two times the length of the shorter leg, and the length of

the longer leg is √__

3 times the length of the shorter leg.

Problem SetDetermine the measure of the indicated interior angle.

1. A

B C

2. D30°

E G F

m�ABC � 60º m�DFE �

3. H

J A K

30° 4. R

60°

S A T

m�HAK � m�TRA �


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Given the length of the short leg of a 30°– 60°– 90° triangle, determine the lengths of the long leg and hypotenuse. Write your answers as radicals in simplest form.

5. 3 ft

c

b30°

60° 6.

5 in.c

b30°

60°

a � 3 ft

b � 3 √__

3 ft

c � 2(3) � 6 ft

7.

6 mmc

b30°

60° 8.

15 cmc

b30°

60°

Given the length of the hypotenuse of a 30°– 60°– 90° triangle, determine the lengths of the two legs. Write your answers as radicals in simplest form.

9.

a20 m

b30°

60° 10.

a16 km

b

30°

60°

c � 20 m

a � 20 ___ 2

� 10 m

b � 10 √__

3 m

11.

a6 3 yd

b30°

60° 12.

a4 2 ft

b30°

60°


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Given the length of the long side of a 30°– 60°– 90° triangle, determine the lengths of the short leg and hypotenuse. Write your answers as radicals in simplest form.

13.

ac

8 3 in.

30°

60° 14.

ac

30°

60°

11 3 m

b � 8 √__

3 in.

a � 8 √__

3 ____ √

__ 3 � 8 in.

c � 2(8) � 16 in.

15.

ac

12 mi30°

60° 16.

ac

18 ft30°

60°

Determine the area of each 30°– 60°– 90° triangle. Round your answer to the nearest tenth, if necessary.

17.

a6 cm

b30°

60°

a � 6 __ 2

� 3 cm

b � 3 √__

3 cm

A � 1 __ 2

� 3 � 3 √__

3

A � 9 √__

3 ____ 2 � 7.8 cm2

The area of the triangle is approximately 7.8 square centimeters.


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18.

a12 km

b30°

60°

19. Universal Sporting Goods sells pennants in the shape of 30º– 60º– 90º triangles.

The length of the longest side of each pennant is 16 inches.

20. A factory produces solid drafting triangles in the shape of 30º– 60º– 90º triangles.

The length of the side opposite the right angle is 15 centimeters.


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Construct each triangle described using the given segment.

21. Construct a 30°– 60°– 90° triangle by first constructing an equilateral triangle with ____

MN as a side, and then bisecting one of the sides.

M N

M N

22. Construct a 30°– 60°– 90° triangle RST by first constructing an equilateral triangle

with ___

RS as a side, and then bisecting the angle at R.

R S


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23. Construct a 30°– 60°– 90° triangle EFG with ___

EF as the side opposite the 30° angle

by first constructing an equilateral triangle.

E F

24. Construct a 30°– 60°– 90° triangle ABC by first copying angle A, and then drawing ___

AB as the hypotenuse.

A

A30°

B


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Name _____________________________________________ Date ____________________

Pasta Anyone? Triangle Inequality Theorem

VocabularyIdentify an example of each term in the diagram of triangle ABC.

1. Triangle Inequality Theorem B

CDA 2. auxiliary line

Problem SetWithout measuring the angles, list the angles of each triangle in order from least to greatest measure.

1.

F

G

H8 in.

9 in.

11 in.

2.

X W

Y

4.7 cm3.6 cm

2.1 cm

The smallest angle of a triangle is opposite the shortest side. So, the angles from least to greatest are �H, �F, �G.


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3. Q

RP

8 ft 4 ft

6.43 ft

4. T S

U

9 in

12 in

15 in

5. F

E G6 yd

9.2 yd

4.6 yd

6. K

M

L

4.2 m

5.2 m

5.8 m

Determine whether it is possible to form a triangle using each set of segments with the given measurements. Explain your reasoning.

7. 3 in., 2.9 in., 5 in. 8. 8 ft, 9 ft, 11 ft

Yes. Because 3 � 2.9 � 5.9, and5.9 is greater than 5.

9. 4 m, 5.1 m, 12.5 m 10. 7.4 cm, 8.1 cm, 9.8 cm

11. 10 yd, 5 yd, 21 yd 12. 13.8 km, 6.3 km, 7.5 km

13. 112 mm, 300 mm, 190 mm 14. 20.2 in., 11 in., 8.2 in.


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15. 30 cm, 12 cm, 17 cm 16. 8 ft, 8 ft, 8 ft

Write an inequality that expresses the possible lengths of the unknown side of each triangle.

17. What could be the length of ___

AB ? 18. What could be the length of ___

DE ?

A

10 m

B 8 m C

D

6 cm

F 9 cm E

AB � AC � BC AB � 10 m � 8 m AB � 18 m

19. What could be the length of ___

HI ? 20. What could be the length of ___

J L?

I

20 in.

H 14 in. G

J

12 ft

K 7 ft L


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21. What could be the length of ____

MN ? 22. What could be the length of ___

QR ?

M

3 cm

N 11 cm O

P

9 mm 13 mm

R Q

skills practice for lesson 4 - ms. hollingsworth - homech...exterior angle of a triangle is greater...

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