Developmental mathematics 3rd edition martin gay solutions manual 1

Developmental Mathematics 3rd Edition Martin Gay

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Chapter 6

Section 6.1 Practice Exercises

1. Figure (a) is part of a line with one endpoint, so → it is a ray. It is ray AB or AB .

Figure (b) has two endpoints, so it is a line

segment. It is line segment RS or RS . Figure (c) extends indefinitely in two directions,

are corresponding angles, as are x and d So all of these angles have the same measure:

are adjacent angles, as are

45 = 135 a and c are corresponding angles so 

angles, so mz = mc = 135

Vocabulary, Readiness & Video Check 6.1

mc = ma = 135 c and z are vertical so it is a line. It is line EF or EF . Figure (d) has two rays with a common endpoint, so it is an angle. It is HVT or TVH or V

2. Two other ways to name z are RTS and STR.

3. a. R is an obtuse angle. It measures between

1. A plane is a flat surface that extends indefinitely.

2. A point has no length, no width, and no height.

3. Space extends in all directions indefinitely. 90 and 180 .

b. N

c. M

is a straight angle. It measures 180 is an acute angle. It measures between

4. A line is a set of points extending indefinitely in

two directions.

5. A ray is part of a line with one endpoint.

0 and 90

d. Q is a right angle. It measures 90

4. The complement of a 29 angle is an angle that measures 90 29 = 61 .

5. The supplement of a 67 angle is an angle that measures 180 67 = 113 .

6. a.

my = mADC mBDC = 141 97 = 44

6. An angle is made up of two rays that share a common endpoint. The common endpoint is called the vertex

7. A straight angle measures 180 .

8. A right angle measures 90 .

9. An acute angle measures between 0 and 90 .

10. An obtuse angle measures between 90 and 180

b.

mx = 79 51 = 28

c. Since the measures of x and y are between 0 and 90, they are both acute angles.

7. Since a and the angle marked 109 are vertical angles, they have the same measure; so

ma = 109 .

Since a and b are adjacent angles, their measures have a sum of 180 So

mb = 180 109 = 71 .

Since b and c are vertical angles, they have the same measure; so mc = 71

11. Parallel lines never meet and intersecting lines meet at a point.

12. Two intersecting lines are perpendicular if they form right angles when they intersect.

13. An angle can be measured in degrees.

14. A line that intersects two or more lines at different points is called a transversal

15. When two lines intersect, four angles are formed. The angles that are opposite each other are called vertical angles.

16. Two angles that share a common side are called adjacent angles.

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17. WUV, VUW, U, x

30. 20 + 70 = 90 , so CAD and DAE are complementary. 63 + 27 = 90 , so BAC and

18. straight angle; 180 EAF are complementary.

19. 180 17 = 163

20. intersect

Exercise Set 6.1

2. The figure has two rays with a common end point. It is an angle, which can be named GHI, IHG, or H

4. The figure has one end point and extends indefinitely in one direction, so it is a ray. It is

32. 38 + 142 = 180 , so there are 4 pairs of supplementary angles: NMX and NMY , NMX and XMZ, YMZ and NMY , YMZ and XMZ

34.

36.

mx = 150 48 = 102

mx = 36 +12 = 48

x and the angle marked 165 are supplementary, so mx = 180 165 = 15 ray ST or → ST

38.

y and the angle marked 165 are vertical angles, so my = 165

6. The figure extends indefinitely in two directions, x and z are vertical angles, so

so it is a line. It is line AB, line t, or AB

8. The figure has two end points, so it is a line segment. It is line segment PQ or PQ

mz = mx = 15

x and the angle marked 44 are supplementary, so mx = 180 44 = 136

y and the angle marked 44 are vertical

10. Two other ways to name w are APC and angles, so my = 44 .

CPA

12. Two other ways to name y are MPQ and QPM

14. H is an acute angle. It measures between 0 and 90

42.

x and z are vertical angles, so

mz = mx = 136

x and the angle marked 110 are vertical angles, so mx = 110

x and y are corresponding angles, so

my = mx = 110

T is an obtuse angle. It measures between 90 and 180

M is a straight angle. It measures 180

44.

mz = my = 110 .

y and z are vertical angles, so 16. 18. 20.

x and the angle marked 39 are supplementary, so mx = 180 39 = 141

y and the angle marked 39 are corresponding angles, so my = 39 22. The complement of an angle that measures 77 is y and z are supplementary, so an angle that measures 90 77 = 13

N is a right angle. It measures 90

mz = 180 my = 180 39 = 141

Chapter 6: Geometry ISM: Developmental Mathematics

24. The supplement of an angle that measures 77 is an angle that measures 180 77 = 103

26. The complement of an angle that measures 22 is an angle that measures 90 22 = 68

28. The supplement of an angle that measures 130 is an angle that measures 180

y can also be named CBD or DBC.

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66. The complement of an angle that measures 53.13 is an angle that measures

= 36.87

68. False; answers may vary

70. True, since 5 + 175 = 180

72. The sides of a rectangle are parallel. Thus, AB and CD, as well as AC and BD are parallel.

74. answers may vary

76. no; answers may vary

3. The radius is half the diameter.

2 = 16 in.

2 = 8 in.

Vocabulary, Readiness & Video Check 6.2

1. Because the sum of the measures of the angles of a triangle equals 180, each angle in an equilateral triangle must measure 60 .

2. rectangular sold; shoeboxes, cereal boxes

Exercise Set 6.2

2. The figure has eight sides, so it is an octagon.

4. The figure has three sides, so it is a triangle.

6. The figure has three sides, so it is a triangle.

8. The figure has seven sides, so it is a heptagon.

10. No two sides of the triangle have the same length, so the triangle is scalene.

12. Two sides of the triangle have the same length. Also, one of the angles is a right angle. Therefore the triangle is an isosceles right triangle.

14. All three sides of the triangle have the same length, so the triangle is equilateral.

4. The diameter is twice the radius. d = 2 r = 2 7 mi = 14 mi

Chapter 6: Geometry ISM: Developmental Mathematics

22. A rectangle with all four sides equal is a square.

24. Half the diameter of a circle is its radius.

26. A quadrilateral with exactly one pair of opposite sides parallel is a trapezoid.

28. A triangle with no equal sides is a scalene triangle.

30. d = 2  r = 2  3 = 6 mi

32. r = d  2 = 13  2 = 6.5 ft

34. d = 2  r = 2  7.8 = 15.6 in.

36. r = d  2 = 2.6  2 = 1.3 m

38. The solid is a cube.

40. The solid is a pyramid.

42. The solid is a sphere.

44. The object has the shape of a cone.

46. The object has the shape of a cylinder.

48. The object has the shape of a rectangular solid.

50. The object has the shape of a rectangular solid.

52. d = 2  r = 2  5.8 = 11.6 m

54. r = d  2 = 78  2 = 39 cm

56. d = 2  r  2  15 = 30 in.

58. 4(87) = 348

60. 2(7.8) + 2(9.6) = 15.6 + 19.2 = 34.8

62. True; since all four sides of a square have the same length and all four angles are right angles, a square is also a regular polygon.

64. False; although two of the sides of a trapezoid are parallel, the other two need not be.

66. True; since opposite sides of a rhombus are parallel, a rhombus is also a parallelogram.

68. answers may vary

Section 6.3 Practice Exercises

1. a. Perimeter = 12 m +12 m +15 m +15 m = 54 meters

6. The unmarked horizontal side has length

20 m 15 m = 5 m. The unmarked vertical side has length 26 m 7 m = 19 m.

P = 15 m + 26 m + 20 m + 7 m + 5 m +19 m

= 92 meters

7. P = 2 l + 2 w

= 2 120 feet + 2 60 feet

= 240 feet +120 feet

= 360 feet

cost = $1.90 per foot 360 feet = $684 The cost of the fencing is $684.

8. a. C =  d =  20 yd = 20 yd  62.8 yd The exact circumference of the watered region is 20 yards, which is approximately 62.8 yards.

b. C =  d =  12 in. = 12 in.  37.68 in. The exact circumference of the clock face is 12 inches, which is approximately 37.68 inches.

Vocabulary, Readiness & Video Check 6.3

1. The perimeter of a polygon is the sum of the lengths of its sides.

2. The distance around a circle is called the circumference

3. The exact ratio of circumference to diameter is

4. The diameter of a circle is double its radius

b. Perimeter = 80 ft + 80 ft + 60 ft + 60 ft 5. Both 22 (or 3.14) and

22  are = 280 feet

2. P = 2 l + 2 w

= 2 22 cm + 2 10 cm

= 44 cm + 20 cm = 64 centimeters

3. P = 4 s = 4 5 feet = 20 feet

4. P = a + b + c

= 5 cm +10 cm + 6 cm

= 21 centimeters

5. Perimeter = 4 km + 6 km + 4 km + 9 km = 23 kilometers

6. The radius of a circle is half its diameter

7. Opposite sides of a rectangle have the same measure, so we can just find the sum of the measures of all four sides.

8. the perimeter of a circle

Exercise Set 6.3

= 38 m

The perimeter is 38 meters.

4. P = 2 yd + 3 yd + 2 yd + 3 yd = 10 yd

The perimeter is 10 yards.

6. P = a + b + c

= 5 units +10 units +11 units

= 26 units

The perimeter is 26 units.

8. Sum the lengths of the sides.

P = 10 m + 4 m +10 m + 9 m + 20 m +13 m = 66 m

The perimeter is 66 meters.

10. All sides of a regular polygon have the same length, so the perimeter is the number of sides multiplied by the length of a side.

P = 4  50 m = 200 m

The perimeter is 200 meters.

12. All sides of a regular polygon have the same length, so the perimeter is the number of sides multiplied by the length of a side.

P = 6 15 yd = 90 yd

The perimeter is 90 yards.

14. P = a + b + c = 8 in. +12 in. +10 in. = 30 in.

The perimeter is 30 inches.

16. P = 4 s = 4 90 ft = 360 ft The player will run 360 feet.

18. P = 8  12 in. = 96 in. The perimeter is 96 inches.

20. P = 2 l + 2 w = 2 70 ft + 2 21 ft = 140 ft + 42 ft = 182 ft

182 feet of fencing is needed.

22. The amount of fencing needed is 182 feet.

182 feet $2 per foot = $364

The total cost of the fencing is $364.

24. All sides of a regular polygon have the same length, so the perimeter is the number of sides multiplied by the length of a side.

P = 5  14 m = 70 m

The perimeter is 70 meters.

26. P = 4 s = 4 3 in. = 12 in. The perimeter is 12 inches.

28. P = 2 l + 2 w

= 2 85 ft + 2 70 ft

= 170 ft +140 ft

= 310 ft

310 ft $2.36 per foot = $731.60

The cost is $731.60.

30. The unmarked vertical side has length

13 in. 6 in. = 7 in.

The unmarked horizontal side has length

30 in. 13 in. = 17 in.

P = 13 in. +13 in. + 30 in.+ 7 in. +17 in. + 6 in.

= 86 in.

The perimeter is 86 inches.

32. The unmarked vertical side has length (2 + 4 + 3) cm = 9 cm.

The width of the figure in the vertical section of height 4 cm is 16 11 = 5 cm. So the unmarked horizontal side has length (9 5) cm = 4 cm.

P = (16 + 2 +11+ 4 + 4 + 3 + 9 + 9) cm = 58 cm

The perimeter is 58 centimeters.

34. The unmarked vertical side has length (22 + 5) km = 27 km.

The unmarked horizontal side has length (12 + 6) km = 18 km

P = (18 + 27 + 6 + 5 + 12 + 22) km = 90 km

The perimeter is 90 kilometers.

2.5 in. = 5 in.  15.7 in. The circumference is exactly 5 inches or approximately 15.7 inches.

38. C =  d =  50 ft = 50 ft  157 ft

The circumference is exactly 50 feet or approximately 157 feet.

40. C = 2  r = 2  10 yd = 20 yd  62.8 yd

The circumference is exactly 20 yards or approximately 62.8 yards.

42. C =  d =  150 ft = 150 ft  471 feet

The circumference of the barn is 150 feet, or about 471 feet.

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44. C =

The distance around is about 17 2 inches or 7 17.29 inches.

46. P = (4.5 + 7 + 9) yd = 20.5 yd

The perimeter is 20.5 yards.

48. C =  d =  11 m = 11 m  34.54 m

The circumference is 11 meters  34.54 meters.

50. P = 4 19 km = 76 km

The perimeter is 76 kilometers.

52. The unmarked vertical side has length (40 9) mi = 31 mi.

The unmarked horizontal side has length (44 9) mi = 35 mi.

P = (44 + 31 + 9 + 9 + 35 + 40) mi = 168 mi

The perimeter is 168 miles.

54. 25 3 7 = 25 21 = 4

56. 6 (8 + 2) = 6 10 = 60

58. (72  2) 6 = 36 6 = 216

66. a. Smaller circle:

62. a. The first age category that 12-year-old children fit into is “Under 13,” so the maximum width is 60 yards and the maximum length is 110 yards.

b. P = 2 l + 2 w

= 2 110 yd + 2 60 yd

= 220 yd +120 yd

= 340 yd

The perimeter of the field is 340 yards.

64. The circle’s circumference is   7 in.  21.98 in.

The square’s perimeter is 4  7 in. = 28 in.

So the square has the greater distance around; b.

16

= 16

 50.24 m Larger circle:

C =   d =   32 m = 32 m  100.48 m

b. Yes, when the diameter of a circle is doubled, the circumference is also doubled.

68. answers may vary

70. The length of the curved section at the top is half of the circumference of a circle of diameter 6 meters.

1 C = 1

d = 1

6 m = 3 m  9.4 meters 2 2 2

The total length of the straight sides is 3 6 m = 18 m.

The perimeter is the sum of these.

9.4 m + 18 m = 27.4 m

The perimeter of the figure is 27.4 meters.

72. The three linear sides have lengths of 7 feet, 5 feet, and 7 feet. The length of the curved side is half of the circumference of a circle with diameter 5 feet, or

1  d = 1  5 = 2.5  7.9 feet

2 2

7 ft + 5 ft + 7 ft + 7.9 ft = 26.9 ft

The perimeter of the window is 26.9 feet.

74. P = 4 s, so s = P  4. s = 18 in.  4 = 4.5 in.

The length of a side is 4.5 inches.

1. A = 1 b h

2

= 1 8 in. 6 1 in.

2 4

= 1 8 in. 25 in.

2 4

= 25 sq in.

The area is 25 square inches.

2. A = s s = 4.2 yd 4.2 yd = 17.64 sq yd

The area is 17.64 square yards.

3. Split the rectangle into two pieces, a top rectangle with dimensions 12 m by 24 m, and a bottom rectangle with dimensions 6 m by 18 m. The area of the figure is the sum of the areas of

Section 6.4 Practice Exercises

Chapter 6: Geometry ISM: Developmental Mathematics

A = 12 m  24 m + 6 m 18 m

= 288 sq m +108 sq m = 396 sq m

The area is 396 square meters.

 153.86 sq cm

The area is 49 square centimeters, which is approximately 153.86 square centimeters.

Vocabulary, Readiness & Video Check 6.4

1. The formula for the area of a rectangle; we split the L-shaped figure into two rectangles, used the area formula twice to find the area of each, and then added these two areas.

ISM: Developmental Mathematics

18. A = b h = 6 m

 4 m = 24 sq m

20. The area of the trapezoid that forms the top of the figure is

1 (b + B)  h =

1 (6 km +10 km)  4 km

2

2 = 32 sq k m. The area of the rectangle that forms the bottom of the figure is l w = 10 km 5 km = 50 sq km. The area of the figure is the sum of these.

A = 32 sq km + 50 sq km = 82 sq km

22. The figure can be divided into two rectangles, one measuring 25 cm by 12 cm and one measuring 20 cm by 3 cm. The area of the figure is the sum of the areas of the two rectangles.

A = 25 cm

12

cm + 20 cm 3 cm

Chapter 6: Geometry

= 300 sq cm + 60 sq cm

= 360 sq cm

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Chapter 6: Geometry ISM: Developmental Mathematics

24. The top of the figure is a square with sides of length 5 in., so its area is

s2 = (5 in.)2 = 25 sq in.

The bottom of the figure is a parallelogram with area b  h = 5 in.  4 in. = 20 sq in.

The area of the figure is the sum of these.

A = 25 sq in. + 20 sq in. = 45 sq in.

26. r = d  2 = 5 m  2 = 2.5 m

A = r 2

=  (2.5 m)2 = 6.25 sq m

 19.625 sq m

28. A = l w = 35 ft 24 ft = 840 sq ft 840 feet of insulation are needed.

30. A = l  w = 28.0 m  6.2 m = 173.6 sq m

The area of the sign is 173.6 square meters.

The house is a rectangle, with area

l  w = 48 ft  24 ft = 1152 sq ft.

The area of the yard is the difference between these.

A = 5472 sq ft 1152 sq ft = 4320 sq ft

The area of the yard is 4320 square feet.

42. a. The top part of the wall is a triangle, with area 1  b  h = 1  24 ft  4 ft = 48 sq ft.

2 2

The bottom part is a rectangle, with area

l

 w = 24 ft  8 ft = 192 sq ft.

The area of the wall is the sum of these.

A = 48 sq ft + 192 sq ft = 240 sq ft

The total area is 240 square feet.

b. Divide the area of the wall by the side area of each brick.

240

 1 = 240 6 = 1440

6 1

The number of bricks needed to brick the

32.

A = s 2 = (54 in.)2 = 2916 sq in.

The area of the top of the pizza is 2916 square inches.

end of the building is 1440.

44. P = 2

34.

A = l w

46. Sum the lengths of the sides.

The area of the double roll is 225 1 2

= 2 ft

square feet.

A = r 2 = (2 ft)2 = 4 sq ft  12.56 sq ft

There are approximately 12.56 square feet of mat.

38. A = l w = 27.5 cm 20.5 cm = 563.75 sq cm

The area of a page is 563.75 square centimeters.

40. Including the house, the land is in the shape of a trapezoid, whose area is 1 (b + B) h = 1 (96 ft +132 ft) 48 ft 2 2

48. P = 3 3 in. = 9 in.

50. Grass seed is planted throughout the surface of a yard, so the situation involves area.

52. Gutters go on the edges of a house, so the situation involves perimeter.

54. Baseboards go around the edges of a room, so the situation involves perimeter.

= 1  228 ft  48 ft 2

= 5472 sq ft.

ISM: Developmental Mathematics

56. Fertilizer is spread throughout the surface of ayard, so the situation involves area.

58. This could be answered by dividing Utah into two rectangles, in two different ways. But it’s easier to regard the state as a 350 mile by 270 mile rectangle from which a 70 mile by 105 milerectangle has been removed.

A = 350 mi  270 mi 70 mi 105 mi

= 94,500 sq mi 7350 sq mi

= 87,150 sq mi

The area of Utah is approximately 87,150 squaremiles.

60. answers may vary

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62. A = l w = 15 ft 10 ft = 150 sq ft

Since 1 yard is 3 feet, 1 square yard is 3 ft 3 ft = 9 square feet.

64.

cost is approximately $108.33.

The area of the circular base of the bowl was

66. The semicircle at the top has radius 2.5 feet, so

3.

The rectangle at the bottom has area

l

w = 7 ft

5 ft = 35 sq ft.

The total area is 3.125 sq ft + 35 sq ft  44.8 sq ft.

68. a. The first age category that 11-year-old children fit into is “Under 12,” so the maximum width is 55 yards and the maximum length is 105 yards.

b. A = l w = 105 yd 55 yd = 5775 sq yd The area of the field is 5775 square yards.

Section 6.5 Practice Exercises

1. V = l  w  h = 5 ft  2 ft  4 ft = 40 cubic ft

The volume of the rectangular box is 40 cubic feet.

The volume of the cylinder is 175 cubic inches, approximately 549.5 cubic inches.

4. V = 1  s2  h

15.3 cu m

The volume of the pyramid is 15.3 cubic meters.

Vocabulary, Readiness & Video Check 6.5

1. The measure of the amount of space inside a solid is its volume

2. Area measures the amount of surface of a region.

3. Volume is measured in cubic units.

4. Area is measured in square units.

5. The perimeter of a polygon is the sum of the

Copyright © 2015 Pearson Education, Inc.

lengths of its sides.

6. Perimeter is measured in units

7. Exact answers are in terms of , and approximate answers use an approximation for

Chapter 6: Geometry ISM: Developmental Mathematics

r = 1 d = 1 54 ft = 27 ft

26.

2 V = 4 r 3 3 = 4 (27 ft)3 3 = 26,244 cu ft

28.

V = 4   r 3 3 = 4  (3 in.)3 3 = 36 cu in.

 113.04 cu in.

The volume is 36 cubic inches  113.04 cubic inches.

30. V = 1  r2 h 3

= 1 (3 km)2(3.5 km) 3 = 10.5 cu km

 

46. r = d = 55 ft = 27.5 ft

2

3 2 3 = 43,534.79

48. First solid:

V = lwh = (6 ft)(2 ft)(4 ft) = 48 cu ft

SA = 2lh + 2wh + 2lw

= 2(6 ft)(4 ft) + 2(2 ft)(4 ft) + 2(6 ft)(2 ft)

= 88 sq ft

Second solid:

V = lwh = (4 ft)(3 ft)(4 ft) = 48 cu ft

SA = 2lh + 2wh + 2lw

= 2(4 ft)(4 ft) + 2(3 ft)(4 ft) + 2(4 ft)(3 ft)

= 80 sq ft no; answers may vary

50. V =  r2 h = (16.3 ft)2  32 ft

= 8502.08 cu ft

 26,696.5 cu ft

The volume of the tank is about 26,696.5 cubic feet. 32. V =

The volume is 10.5 cubic kilometers  33 cubic kilometers.

1 s 2 h = 1 (7 in.)2 1 163 cu in.

3 3 The volume of the toy is 1 3

163 cubic inches.

34. V = s3 = (2 in.)3 = 8 cu in.

1. The supplement is 180 27 = 153 The complement is 90 27 = 63 .

2. x and the angle marked 105 are adjacent

36. answers may vary angles, so mx = 180 105 = 75 . y and the angle marked 105 are vertical angles, so

my = 105 z and x are vertical angles, so

38. 72 = 7 7 = 49

mz = 75 40. 20

2 = 20 20 = 400

angles, so mx = 180 52 = 128 . y and the

Developmental Mathematics

Chapter 6: Geometry

4. The sum of the interior angles of a triangle is

10. P = 11 mi + 5 mi + 11 mi + 5 mi = 32 mi A = b h = 11 mi 4 mi = 44 sq mi

11. The unmarked vertical side has length

cm + 7 cm = 10 cm. The unmarked horizontal side has length

5. Recallthat49 = 7and64 = 8.Since62isbetween 49 and 64, then 62 is between 49 and 64. Thus, 62 is between 7 and 8. Since

7. hypotenuse = = (7)2 + (9)2 49 + 81

3. The reverse process of squaring a number is finding a square root of a number. = 130

 11

The hypotenuse is 130 kilometers, or approximately 11 kilometers.

8. leg = = =

(11)2 (7)2 121 49 72

The length of the leg is 72 feet, or approximately 8.49 feet.

9. 100 yd

4. The numbers 9, 1, and 1 are called perfect

5. Label the parts of the right triangle.

6. The Pythagorean theorem can be used for right triangles.

7. = 7 because 72 or 7 7 = 49.

8. Since 15 is between 9 = 3 and 16 = 4, and 15 is much closer to 16 than to 9, we know that is closer to 4 than to 3.

9. The Pythagorean theorem works only for right triangles. Exercise Set 6.6 12,809 = c 113  c

The diagonal is approximately 113 yards.

2. = 3 because 32 = 9.

4. = 12 because 122 = 144. 6. =

because

.

Chapter 6: Geometry ISM: Developmental Mathematics

hypotenuse = (leg)2 + (other leg)2

52. hypotenuse = (leg)2 + (other leg)2 =

 33.541

The hypotenuse has length of about 33.541 units.

46. 21 21

hypotenuse = (leg)2 + (other leg)2

 1.414

The length of the diagonal is about 1.414 miles.

54. leg = =

(hypotenuse)2 (leg)2 1682 602 28,224 3600 24,624

 156.9

56. hypotenuse = =

(leg)2 + (other leg)2

The height of the antenna is about 156.9 feet. = 212 + 212 = 441+ 441 = 882

702 +1102 4900 +12,100

 29.698 = 17,000

The hypotenuse has length of about 29.698 units.

48. 8

 130.4

The length of the diagonal is about 130.4 yards.

leg = (hypotenuse)2 (other leg)2

The leg has length of about 4.123 units.

hypotenuse = (leg)2 + (other leg)2 = 122 + (22.5)2 = 144 + 506.25

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64. Recall that 25 = 5 and 36 = 6. Since 27 is

between 25 and 36, then 27 is between 25

and 36. Thus, 27 is closer to 25, then is between 5 and 6. Since is closer to 25, or

5.Check:  5.20

66. Recall that 81 = 9 and between 81 and 100, then = 10. Since 85 is is between

and 100. Thus, 85 is between 9 and 10.

Since 85 is closer to 81, then 85 is closer to 81, or 9. Check:  9.22

68. answers may vary

+ 452 502

+ 2025 2500 2425  2500 No; the set does not form the lengths of the sides of a right triangle.

Section 6.7 Practice Exercises

1. a. The triangles are congruent by Side-AngleSide.

b. The triangles are not congruent.

2. 9 meters = 9

13 meters 13

The ratio of corresponding sides is 9 . 13

3. a. The ratios of corresponding sides in the two triangles are equal, so:

n = 10

4. The triangle formed by the sun’s rays, Tammy, and her shadow is similar to the triangle formed by the sun’s rays, the building, and its shadow.

Vocabulary, Readiness & Video Check 6.7

1. Two triangles that have the same shape, but not necessarily the same size are congruent. false

2. Two triangles are congruent if they have the same shape and size. true

3. Congruent triangles are also similar. true

4. Similar triangles are also congruent. false

5. For the two similar triangles, the ratio of corresponding sides is 5 false 6

6. A and D are congruent, B and E are

congruent, C a = b = c

and F are congruent, and

Chapter 6: Geometry ISM: Developmental Mathematics

n = 8

The missing length is 8 units.

7. M and Y are congruent, N and X are

congruent, P

and Z are congruent, and

8. Since all sides of both triangles are given, and no angle measures are given, we used SSS.

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Chapter 6: Geometry ISM: Developmental Mathematics

9. The ratios of corresponding sides are the same.

2. The triangles are congruent by Side-Side-Side.

4. The triangles are not congruent.

6. The triangles are congruent by Angle-SideAngle.

8. The triangles are congruent by Side-Angle-Side.

The tree is 365 feet tall.

34. The triangle formed by the sun’s rays, Lloyd, and his shadow is similar to the triangle formed by the sun’s rays, the building, and its shadow.

the height of the building.

Let x be the width of the banner.

The building is about 50 feet tall.

= 10,000 + 2500 = 12,500

The length of the diagonal is approximately 112 yards.

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z = 1

15. When two of the four angles from Exercise 14

6 1 share a common side, they are called adjacent 4 angles.

z 1 = 6 1

4

z = 24

The lengths of the sides of the deck are 12 feet, 18 feet, and 24 feet.

Chapter 6 Vocabulary Check

1. A right triangle is a triangle with a right angle. The side opposite the right angle is called the hypotenuse, and the other two sides are called legs.

2. A line segment is a piece of a line with two endpoints.

3. Two angles that have a sum of 90 are called complementary angles.

4. A line is a set of points extending indefinitely in

two directions.

5. The perimeter of a polygon is the distance around the polygon.

6. An angle is made up of two rays that share the same endpoint. The common endpoint is called the vertex

7. Congruent triangles have the same shape and the same size.

8. Area measures the amount of surface of a region.

9. A ray is a part of a line with one endpoint. A ray extends indefinitely in one direction.

16. An angle whose measure is between 90 and 180 is called an obtuse angle.

17. An angle that measures 90 is called a right angle.

18. An angle whose measure is between 0 and 90 is called an acute angle.

19. Two angles that have a sum of 180 are called supplementary angles.

20. Similar triangles have exactly the same shape but not necessarily the same size.

Chapter 6 Review

1. A is a right angle.

2. B

3. C

is a straight angle. is an acute angle.

4. D is an obtuse angle.

5. The complement of a 25 angle has measure 90 25 = 65

6. The supplement of a 105 angle has measure 180 105 = 75

7.

8.

10. m



x = 105 15 = 90 m

11. 47 + 133 = 180 , so a and b are

supplementary. So are b and c, c and

12. An angle that measures 180 is called a straight angle. d, and d and a

13. The measure of the space of a solid is called its

ISM: Developmental Mathematics

volume

14. When two lines intersect, four angles are formed. The angles that are opposite each other are called vertical angles.

Chapter 6: Geometry

12. 47 + 43 = 90, so x and w are complementary. Also, 58 + 32 = 90 , so y and z are complementary.

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Chapter 6: Geometry ISM: Developmental Mathematics

13. x and the angle marked 100 are vertical

32. The polygon has 6 sides, so it is a hexagon. angles, so mx = 100 x and y are adjacent angles, so my = 180 100 = 80

y and z are vertical angles, so mz = 80

14. x and the angle marked 25 are adjacent angles, so mx = 180 25 = 155 x and

33. The triangle has 3 equal sides, so it is equilateral.

34. The triangle has a right angle and 2 equal sides, so it is an isosceles right triangle.

35. P = 27 m +17 1 m + 27 m +17 1 m = 89 m

y are vertical angles, so my = 155 z and 2 2

the angle marked 25 are vertical angles, so mz = 25 .

36. P = 11 cm + 7.6 cm + 12 cm = 30.6 cm

37. The unmarked vertical side has length

15.

x and the angle marked 53 are vertical angles, so mx = 53 x and y are alternate interior angles, so my = 53 . y and z are adjacent angles, so mz = 180 53 = 127

8 m 5 m = 3 m. The unmarked horizontal side has length 10 m 7 m = 3 m.

P = (7 + 3 + 3 + 5 + 10 + 8) m = 36 m

16.

x and the angle marked 42 are vertical

y are alternate

38. The unmarked vertical side has length 5 ft + 4 ft + 11 ft = 20 ft.

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ISM: Developmental Mathematics

27. The solid is a square-based pyramid.

28. The solid is a rectangular solid.

29. d = 2 r = 2 9 in. = 18 in.

30. r = d  2 = 4.7 m  2 = 2.35 m

31. The polygon has 5 sides, so it is a pentagon.

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Chapter 6: Geometry

51. The unmarked horizontal side has length 13 m 3 m = 10 m. The unmarked vertical side has length 12 m 4 m = 8 m. The area is the sum of the areas of the two rectangles. A = 12 m 10 m + 8 m 3 m = 120 sq m + 24 sq m = 144 sq m

52. A =

w = 36 ft  12 ft = 432 sq ft The area of the driveway is 432 square feet.

53. A = 10 ft  13 ft = 130 sq ft 130 square feet of carpet are needed.

 307.72 cu in. The volume of the can is about 307.72 cubic inches.

60. Find the volume of each drawer.

Chapter 6: Geometry ISM: Developmental Mathematics

65. = 1 10 because 1 1 = 1 . 10 10 100

66. hypotenuse =

68. leg =

(leg)2 + (other leg)2

+ 25

202 + 212

400 + 441

841

29

71. The height is the leg of a right triangle whose hypotenuse is 126 feet and whose other leg is 90 feet.

2 (other leg)2



The height of the building is about 88.2 feet.

8 n = 37.5 2 2 73. n = 20

69. leg = (hypotenuse) (other leg)

= 24,025 15,376

= 8649

= 93

The leg is 93 units.

74. = 5.8 8 8n = 5.8 24 8n = 139.2 n = 139.2 8 n = 17.4 hypotenuse = = = = 8100 + 8100 115 (leg)2 + (other leg)2 902 + 902 1 100

Chapter 6: Geometry ISM: Developmental Mathematics

is similar to the triangleformed by the sun’s rays, the building, and its shadow. Let x be the height of the building, in feet.

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ISM: Developmental Mathematics

76. x = 2 10 24

77. The supplement of a 72 angle is an angle that measures 180

72

78. The complement of a 1

angle is an angle that measures

44

= 78

x and the angle marked 85 are adjacent angles, so

x = 180

85 = 95

82. Let y be the angle corresponding to x at the bottom intersection. Then y and the angle marked 123

are adjacent angles, so

Chapter 6: Geometry

87. The unmarked horizontal side has length 40 ft 22 ft 11 ft = 7 ft.

P

(22 +15 + 7 +15 +11+ 42 + 40 + 42) ft =

88. The unmarked horizontal side has length 43 m 13 m = 30 m. The unmarked vertical side has length 42 m 14 m = 28 m. The area is the sum

the areas of the two rectangles.

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