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GEOMETRY
Chapter Two
True–False. Mark as true any statement that is always true. Mark as false any statement that is
never true or that is not necessarily true. Be able to justify your answers.
1. If triangle ABC and triangle DEF are both 30–60 triangles, then AB must be congruent to
DE .
2. An angle of degree measure 180 is a reflex angle.
3. Postulates are statements we assume to be true without proof.
4. Any three points are contained in exactly one plane.
5. A right triangle has two acute angles.
6. An obtuse triangle has at least two obtuse angles.
7. All regular quadrilaterals have four lines of symmetry.
8. All regular polygons tessellate the plane.
9. The parallel sides in a trapezoid are called the legs of the trapezoid.
10. Regular polygons are polygons with all sides congruent and all angles congruent.
11.
A soccer ball is a regular polyhedron.
12.
The lateral faces of a rectangular prism meet at the apex of the prism.
13.
The two opposite faces of a cone are called the bases of the cone and they can be oval
(elliptically) shaped as well as truly circularly shaped.
14.
A pyramid with rectangular faces is called a rectangular pyramid.
15.
A sphere is a polyhedron.
16. The bases of a right regular prism may be congruent, parallel hexagons.
17. The lateral faces of a right square pyramid are isosceles triangles.
18. If there are 1.057 quarts in one liter and four quarts in one gallon, then there are 4.228 liters in
one gallon.
Multiple Choice. Mark the letter of the single BEST response. Be sure to read all the choices for
each problem before deciding.
19.
The measure of an acute angle is a number, n, such that:
(a) 0 n 90
(b) 0 n 90
(c) 0 n 90
(d) 0 n 90
(e) 0 n 180
20. Which of the following are simple closed curves?
(a) A circle.
(b) A polygon.
(c) A cube.
(d) Both (a) and (b) are correct.
(e) Both (a) and (c) are correct.
Chapter 2
21. Two segments are congruent if
(a) they intersect.
(b) they are parallel.
(c) they have the same length.
(d) they share one endpoint.
(e) None of these are correct.
22. If a triangle has two congruent sides and an angle with measure 90, then it is called
(a) right.
(b) scalene.
(c) isosceles.
(d) acute.
(e) Both (a) and (c) are correct.
23. If all the angles in a quadrilateral are right angles, then the quadrilateral might be
(a) a trapezoid.
(b) a kite.
(c) a rectangle.
(d) a square.
(e) Both (c) and (d) are correct.
24. Assume the pentagon in the star is regular. The measure of A is?
(a) 36
(b) 54
(c) 72
(d) 108
(e) 144
25. An equilateral triangle has how many lines of symmetry?
(a) 0
(b) 1
(c) 2
(d) 3
(e) 4
26. If a set of polygons forms a tessellation in the plane, then
(a) they cover the plane with no gaps.
(b) they are all congruent polygons.
(c) none of the regions overlap.
(d) Both (a) and (c) are correct.
(e) Both (a) and (b) are correct.
27. The measure of a vertex angle in a regular polygon with nine sides is
(a) 140
(b) 220
(c) 1260
(d) 20
(e) 40
28. Two faces of a prism intersect in
(a) a vertex.
(b) an edge.
(c) a base.
(d) a side.
(e) Both (a) and (b) are correct.
11
12
Chapter 2
29. The slant height of a right regular pyramid is
(a) the perpendicular distance from its apex to the base of the pyramid.
(b) the height of its lateral face.
(c) the length of a base edge.
(d) the length of a lateral edge.
(e) None of these is correct.
30. Which of the following are regular polyhedra?
(a) An octahedron.
(b) A soccer ball.
(c) A right circular cylinder.
(d) A sphere.
(e) Both (a) and (b) are correct.
31. If there are 2.54 cm in one inch then, in one foot there are
(a) 30.48 cm.
(b) 304.8 mm.
(c) 0.3048 meters.
(d) Both (a) and (b) are correct.
(e) Answers (a), (b) and (c) are all correct.
Fill in the Blanks. Complete each statement with a word or phrase that makes it true.
32. A line segment containing the center and with endpoints on a circle is called a
.
33. A portion of a line that has one endpoint and extends indefinitely in one direction is called
a(n)
.
34. If the sum of the measures of two angles is 90, then they are called
.
35. The side of a right triangle that is opposite the right angle is called the
.
36. If the sum of the vertex angles of a polygon is 540, then it is called a(n)
.
37. The faces of a right hexagonal prism are
.
38. Name the five Platonic Solids. For each, give the name of the shape of the face and list the
number of faces.
a.
Solid:
Shape of faces:
Number of faces:
b.
Solid:
Shape of faces:
Number of faces:
c.
Solid:
Shape of faces:
Number of faces:
d.
Solid:
Shape of faces:
Number of faces:
e.
Solid:
Shape of faces:
Number of faces:
39. For an oblique regular hexagonal pyramid, the name of the base is:
and the name of the lateral faces is:
Writing. Write your answers concisely and completely. Feel free to use figures and/or tables to
illustrate the points you are making.
40. Compare and contrast the following types of quadrilaterals: trapezoid, parallelogram,
rhombus, rectangle and square.
41. How are sides and angle measures used to classify quadrilaterals?
42. Consider a regular polygon with n sides. Discuss its characteristics, including sides, angles and
symmetries.
Chapter 2
13
43. Which polygons can be used to make a regular tessellation? Explain why.
Exercises and Problems.
44. In the figure, FAB 30, CAB 66, GAD 23, BA
EA and G and F are collinear.
C
B
G
A
F
D
(a)
(b)
(c)
(d)
E
What is the measure of DAE?
Name three pairs of supplementary angles in this figure.
Name all of the pairs of complimentary angles in this figure.
Name two adjacent angles in this figure.
45. Convert each of the following angle measure to degrees and minutes. Round to the nearest
minute if necessary.
(a) 43.7
(b) 113.67
(c) 56.4
(d) 29.11
46. How many different rays have B as their initial point? Name them.
A
B
C
D
E
47. Points A, B, C and D are located on line l. Their corresponding coordinates are: A 2.35,
B .4, C .7 and D 3.25. Find each of the following distances:
A
(a)
(b)
(c)
(d)
B
C
D
AC
BD
BC
AD
48. The measure of X is 12 more than three times Y . If X and Y are supplementary; find
the measure of X and classify it as acute, right, obtuse or straight.
49. On a standard clock, find the measure of the obtuse angle formed by the hour hand and the
minute hand at 3:37 and 30 seconds.
50. In the figure, the measure of APB is 72 and he measure of DPE is 55. What is the
measure of BPD?
B
A
C
P
D
E
14
Chapter 2
51. Given AB on the square lattice.
(a) How many triangles can be drawn that have AB as a side?
(b) How many of these triangles are isosceles triangles?
(c) How many of these triangles are acute triangles?
(d) How many of these triangles are right triangles?
(e) How many of these triangles are obtuse triangles?
B
A
52. Several triangles are shown below with congruent sides, congruent angles and right angles shown.
(a) Which ones are scalene triangles?
(b) Which ones are isosceles triangles?
(c) Which ones are right triangles?
(d) Which ones are equilateral triangles?
(e) Which ones are obtuse triangles?
B
E
A
C
D
F
G
J
L
I
H
K
M
N
Q
P
R
T
53. The measures of two angles of a triangle are 38 and 102.What is the measure of the third angle?
54. Use a protractor and a ruler to draw a quadrilateral with two angles of 120 and 3 congruent
sides. Name the shape that you have created.
55. Use a protractor and a ruler to draw a hexagon where all of the vertex angles are 120 but the
hexagon is not a regular hexagon.
56. Use this diagram of a regular pentagon to explain how to determine the formula for finding
the vertex angle in a regular polygon.
Chapter 2
15
57. In the following figure, each polygon is regular. Find the angle measures X and Y.
X
Y
58. If the vertex angle of a regular polygon has measure 168, how many sides does it have?
59. Given the pyramid below, verify Euler’s Formula: F V E 2.
60. Answer the following questions about the given prism.
C
E
D
A
B
(a)
(b)
(c)
(d)
(e)
F
Name the bases of the prism.
Name the lateral faces of the prism.
Name the faces that are hidden from view.
Name the edges of the prism.
Name the prism by type. Be as specific as possible.
61. When the following net is folded, what is the name of the resulting polyhedron? Give as
complete a description as possible.
62. Use dimensional analysis to convert 0.3 days to minutes.
63. Use dimensional analysis to convert 246 ft/sec to meters per minute. (Remember that one inch
is 2.54 centimeters.)
16
Chapter 2
64. Georgi ran a mile in 4 minutes and 18 seconds.
(a) How fast did she run in miles per hour?
(b) How fast did she run in miles per minute?
(c) How fast did she run in kilometers per hour? (Remember there are 0.6214 miles in one
kilometer)
Applications.
65. A surveyor measures an angle to be 131536. In order to use this measure in his calculations
he must convert it to degrees. Express this angle measure in degrees and a decimal fraction of
one degree.
66. The bearing of a line segment is the acute angle that the line segment makes with a northsouth line. If the bearing of PQ is N 70 E and the bearing of PR is S 32 W, what is the
measure of the obtuse angle; QPR?
67. An airplane takes off from an airport at a bearing along AB , of N 13 E, and a second plane
leaves the same airport at bearing N 30 W along AC . A third plane leaves the airport at S 87
W along AD . What is the measure of the angle between the first and second planes? Between
the first and third planes? (Remember: The bearing of a line segment is the acute angle that
the line segment makes with a north-south line.)
68. In order to complete an edge in a tiling job, a builder needs an isosceles trapezoid with a base
that fits along the side of the regular hexagonal tiles he is using. Can he form the trapezoid in
one cut? If so, how might he do this? If not, why not?
69. Using the given triangle lattice, if possible, sketch the following shapes. If not possible, explain why.
A
B
(a)
(b)
(c)
(d)
S
Sketch four non-congruent non-right parallelograms with AB as one side.
S
Sketch all of the squares with AB as one side.S
Sketch all of the non-square rectangles with AB as one side.
S
Sketch an isosceles trapezoid with AB as one side.
70. For the given figure the hexagon is regular and the shaded regions are rhombi.
(a) Describe the rotational symmetries of the figure or explain why none exist.
(b) Sketch the lines of symmetries on the figure or explain why none exist.
A
Chapter 2
71. For the given figure both triangles are equilateral triangles.
(a) Describe the rotational symmetries
of the figure or explain why none exist.
(b) Sketch the lines of symmetries on the figure
or explain why none exist.
17
A
S
S
72. In the following figure: BF and AG are parallel and DEF is an isosceles triangle.
A
69°
C
102°
B
H
74°
G
F
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
E
List two supplementary angles.
What shape is ABDEGHA?
What is the measure of DFG?
What is the measure of BAH?
How many obtuse interior angles are in the figure? List the angle names and measures.
List one set of vertices that form a pentagon.
List two sets of vertices that form scalene triangles
Which angles are congruent? List all congruent angle sets.
73. A contractor finds that the angle of elevation (the acute angle between the horizontal and the
line of sight) to the top of a building is 4635 as shown below. What is the measure of QRP?
R
46° 35
P
Q
74. A forester from his vantage point on a hillside sees a puff of smoke in a valley below him. He
measures the angle of depression (the acute angle between the horizontal and the line of
sight) to the smoke from his position to be 211315 (see figure). Find the measure of PSF in
degrees, minutes and seconds.
F
P
21° 13 15
S
18
Chapter 2
75. A surveying party traversed the boundary of an area designated as a wetland. The interior
angles formed were 9012, 13246, 10216, 12040 and 94. Determine the error, if any, in this
traverse. (Remember: A traverse is the measurement of a polygonal region, made in a series of
legs so that one leg is the start of the next leg and the beginning of the first leg connects with
the end of the last leg.)
76. A certain mail order outlet uses a container in the shape shown below. Ignoring the flaps used
to secure the sides together, describe and sketch the shape of the box when it is unfolded.
77. A right regular n-gon pyramid has 28 edges.
(a) What n-gon is it? Show your work.
(b) How many faces does the pyramid have?
(c) How many vertices does the pyramid have?
78. A right regular n-gon prism has 64 vertices.
(a) What n-gon is it? Show your work.
(b) How many faces does the prism have?
(c) How many edges does the prism have?
79. A can of beans is in the shape of a right circular cylinder. To recycle the can, the top and
bottom are cut out and the remaining part of the can is cut and laid flat. Describe the shapes of
the pieces of metal that remain for recycling.
80. Water is flowing along a stream at the rate of 1200 gallons per minute. What is this rate in liters
per second? Round to the nearest hundredth. (Remember: There are four quarts in a gallon
and 1.057 quarts in a liter.)
81. A foreign car has a gas tank that holds 45 liters of gasoline. Assuming the tank is empty and
gas is selling for $1.23 a gallon, how much will it cost to fill this tank with gasoline?
(Remember: There are four quarts in a gallon and 1.057 quarts in a liter.)
82. Light travels at 186,282 miles per second, one year is 365 days and Neptune is 2,697,000,000
miles from Earth. If a light year is the distance light travels in one year, how far is it from
Neptune to Earth in light years?
83. If a car gets 32.6 miles to the gallon, what is its gas mileage in kilometers per liter? Round to
the nearest hundredth. (Remember: There are four quarts in a gallon, 1.057 quarts in one liter,
2.54 centimeters in one inch and 5280 feet in one mile.)
GEOMETRY
Chapter Two—Answers
True–False.
1. F: The triangles may be different sizes and AB and DE may be different lengths.
2. F: An angle of degree measure 180 is a straight angle while a reflex angle has measure
180 n 360.
3. T
4. F—If the points are collinear they are contained in infinitely many planes.
5. T
6. F: Two obtuse angles would sum to greater than 180.
7. T
8. F: Only regular triangles, squares and regular hexagons.
9. F—The parallel sides are called bases.
10. T
11. F: It is semiregular.
12. F: Prisms do not have an apex, this is true for a pyramid.
13. T
14. F: All pyramids have triangular faces, not rectangular.
15. F: A sphere is not comprised of polygonal faces.
16. T
17. T
18. F—1 gallon 3.784 liters.
Multiple Choice.
19. d
20.
24. a
25.
29. b
30.
d
d
a
21.
26.
31.
c
d
e
22.
27.
e
a
23.
28.
e
b
Fill in the Blanks.
32. Diameter
33. ray
34. complementary
35. hypotenuse
36. pentagon
37. rectangular regions
38. (a) Solid: Tetrahedron Shape of faces: Triangle Number of faces: 4
(b) Solid: Cube Shape of faces: Square Number of faces: 6
(c) Solid: Octahedron Shape of faces: Triangle Number of faces: 8
(d) Solid: Dodecahedron Shape of faces: Pentagon Number of faces: 12
(e) Solid: Icosahedron Shape of faces: Triangle Number of faces: 20
39. Base is: Regular Hexagon
Lateral Faces Scalene Triangles
Writing.
40. Congruent sides and parallel sides should be mentioned.
41. Squares and rectangles have all right angles
Squares and rhombi have all four
parallelograms
congruent sides
Parallelograms have opposite sides parallel
Trapezoids have exactly one pair of
rectangles squares rhombus
parallel sides and if the two nonparallel
sides are congruent, it is an isosceles
trapezoid
Kites have two pairs of congruent sides.
non-parallelograms
trapezoids
kites
isosceles
trapezoids
94
Chapter 2–Answers
42. Congruent angles, congruent sides, number of angles, sides, and lines of symmetry should be
mentioned.
43. Regular triangles because 60 evenly divides into 360; squares because 90 evenly divides into
360; regular hexagons because 120 evenly divides into 360.
Exercises/Problems
44. (a) 90 30 60, 60 23 83, 180 83 97, DAE 97
(b) GAD and DAF, GAC and CAF, FAB and GAB,
(c) BA and EA are the only complimentary angles in this figure.
(d) There are many, here are five pairs: GAC and CAB, CAB and BAF, BAF and
FAE, FAE and EAD, EAD and DAG
45. (GET CORRECT MINUTE SYMBOL)
(a) 4342
(b) 11340
(c) 5624
(d) 297
!
!
!
!
46. BA, BC, BD and BE
47. (a) 3.05
(b) 3.65
(c) 1.1
(d) 5.6
48. X 124; obtuse.
49. 135
50. 53
51. (a)21 (b) 4 (c) 5 (d) 6 (e) 10
52. (a) DEF, QRT
(b) ABC, JKL, MNP, GHI
(c) DEF, MNP
(d) GIH
(e) JKL, QRT
53. 40
54. An isosceles trapezoid.
55. Any “stretched” out hexagon will work.
56. Divide the pentagon into three triangles. The sum of the degree measures
of the three triangles is the same as the sum of the degree measures of the
vertex angles of the regular pentagon. Since each vertex angle in the regular
180° * 3
. Since 3 n 2 for
5
n 5; in general you can see that for a regular n-gon, the measure
pentagon is the same, 5v 180 3; v of the vertex angle is v 180° * (n - 2)
.
n
Chapter 2–Answers
57.
58.
59.
60.
61.
62.
63.
64.
X 48, Y 32
30
F 6, V 6, E 10, so F V E 2 Q 6 6 10 2 Q 12 12
(a) ABC, DEF
(b) ABFD, BFEC, ACED
(c) DEF, ACED, ABFD
(d) CE, EF, BF, CB, AC, AB, AD, DE, DF
(e) right triangular prism
A right regular pentagonal prism.
432 minutes
4498.848 m/min
(a) 13.95 miles per hour
(b) .23 miles per minute
(c) 22.45 kilometers per hour
Applications.
65. 13.26°
66. 142
67. 43, 106
68.
69.
(a)
There are many, here are some examples
A
B
(b)
There is only one such square
A
B
95
96
Chapter 2–Answers
(c)
This is not possible, the only way to connect lines, on the grid, forming right angles at A
and B, is to form a square.
A
B
(d)
There is only one possible isosceles trapezoid that fits on this triangular grid
A
B
70. (a) There are two nontrivial rotational symmetries about point A. 120 and 240
(b) There are no lines of symmetry—there is no way to sketch a line that you can fold the
figure in half around and match the opposite sides.
71. (a) There are five nontrivial rotational symmetries about point A. 60, 120, 180, 240
and 300
(b) There are six lines of symmetry as shown here:
A
72. (a) CHD & GHD, CHD & AHC, AHC & AHG, AHG & DHG
(b) A hexagon
(c) 127
(d) 41
(e) AHC 102, GHD 102 and BCH 152
(f) Example: ABDEGA
(g) Example: AHG and GHE
(h) DFE and DEF, DHG and CHA, CHD and AHG,
73. 43 25
74. 68 46 45
75. It is short 6 feet.
Chapter 2–Answers
97
76. Three rectangles and 2 triangles
77. N-gon pyramids have 2n edges; n 14, n 1 vertices, V 15, n 1 faces F 15. Check with
Euler’s formula: 15 15 F V E 2 30.
(a) n 14
(b) F 15
(c) V 15
78. N-gon prisms have 2n vertices; n 32, n 2 faces, F 34, 3n edges E 96. Check with Euler’s
formula: 34 64 F V 96 2 98.
(a) n 32
(b) F 34
(c) E 96
79. Two circles and one rectangle
80.
81.
82.
83.
75.69 1/sec
$14.63
2178 light years
13.86 km/l