Download GLCE/HSCE: Geometry Assessment

GLCE/HSCE: Geometry Assessments
Unit 1
L4.1.1 Distinguish between inductive and deductive reasoning, identifying and providing
examples of each.
1. The process of using facts, rules, definitions, or properties in logical order to reach a
conclusion is called
A.
B.
C.
D.
conjecturing
inductive reasoning
deductive reasoning
detachment reasoning
Answer: C
2. Which of the following is an example of inductive reasoning?
A. Two altitudes of a right triangle are perpendicular to each other. Jamie drew a triangle
with two perpendicular altitudes. Therefore, she drew a right angle.
B. The sum of the interior angles of a triangle is 180°. Leroy drew a triangle where the
sum of two of the angles was 100°. Leroy concluded that the third angle of his
triangle was 80°.
C. The diagonals of a square bisect its right-angle vertices. Kelly drew a square with
diagonals. Therefore, the angles formed at each vertex are complementary.
D. Squares, rectangles, and rhombuses have four sides and are classified as
quadrilaterals. Chan drew a four-sided figure. Chan concluded that his figure is a
quadrilateral.
Answer: C
L4.1.2 Differentiate between statistical arguments (statements verified empirically using
examples or data) and logical arguments based on the rules of logic.
1. Assuming the following statements are true, which of the following is a valid conclusion?
Some musicians are happy people.
All happy people like music.
A.
B.
C.
D.
Some musicians like music.
Some happy people do not like music.
All musicians like music.
All happy people are musicians.
Answer: A
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2. Which of the following groups of statements represents a valid argument?
A. Given: All four sided figures are quadrilaterals. All parallelograms are quadrilaterals.
Conclusion: All parallelograms are quadrilaterals.
B. Given: All rectangles have angles. All squares have four sides.
Conclusion: All rectangles are squares.
C. Given: All quadrilaterals have four sides. All squares have four sides.
Conclusion: All quadrilaterals are squares.
D. Given: All squares have congruent sides. All rhombuses have congruent sides.
Conclusion: All rhombuses are squares.
Answer: A
3. Daniel wants to send Jasmine flowers for her birthday. At the flower store, he can choose
between roses, irises, or carnations. The salesperson tells Daniel that 50% of the customers
buy roses, 30% buy carnations, and 20% buy irises. Which of the following is a valid
conjecture?
A.
B.
C.
D.
More customers buy roses than carnations.
The salesperson likes carnations.
Jasmine will be excited to receive flowers for her birthday.
Daniel will buy Jasmine irises.
Answer: A
L4.1.3 Define and explain the roles of axioms (postulates), definitions, theorems,
counterexamples, and proofs in the logical structure of mathematics. Identify and give
examples of each.
1. A _________ is a statement that describes a fundamental relationship between the basic
terms of geometry and is accepted as true.
A.
B.
C.
D.
theorem
proof
definition
postulate
Answer: D
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2. In the figure, points A, B, and C lie in plane Z. Which of the following postulates can be used
to show that A and B are collinear?
A.
B.
C.
D.
If two planes intersect, then their intersection is a line.
If two lines intersect, then their intersection is exactly one point.
Through any three points not on the same line, there is exactly one plane.
Through any two points, there is exactly one line.
Answer: D
3. State the counterexample that demonstrates that the converse of the following statement is
false:
If an angle measures 48°, then it is acute.
A.
B.
C.
D.
An angle measures 56° and is acute.
All acute angles have measures between 0° and 90°.
An angle measures 48° if and only if it is acute.
If an angle is acute, then it measures 48°.
Answer: B
L4.3.3 Explain the difference between a necessary and a sufficient condition within the
statement of a theorem. Determine the correct conclusions based on interpreting a theorem
in which necessary or sufficient conditions in the theorem or hypothesis are satisfied.
1. The quadrilateral ABCD is a parallelogram.
Which of the following pieces of information would suffice to prove that ABCD is a
rectangle?
A.
B.
C.
D.
AB = AD
angle A and angle B are supplementary
measure of angle B = measure of angle D
AC = BD
Answer: C
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2. Which is a necessary and sufficient condition for a parallelogram to be classified as a
rectangle?
A.
B.
C.
D.
Opposite sides are congruent.
Opposite sides are parallel.
Diagonals bisect each other.
All angles are right angles.
Answer: D
G1.1.1 Solve multistep problems and construct proofs involving vertical angles, linear pairs
of angles, supplementary angles, complementary angles, and right angles.
1. The measures of some angles are given in the figure.
What is the value of x?
A.
B.
C.
D.
70
85
80
65
Answer: A
2. George used a decorative fencing to enclose his deck.
Using the information on the diagram and assuming the top and bottom are parallel, the
measure of angle x is
A.
B.
C.
D.
80°
130°
50°
100°
Answer: C
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3. The Department of Transportation wants to extend the intersecting road across the highway,
as indicated by the dotted line.
What should x be to ensure that the intersecting road and the new construction form a straight
line?
A.
B.
C.
D.
55°
125°
35°
105°
Answer: A
G1.1.3 Perform and justify constructions, including midpoint of a line segment and bisector
of an angle, using straightedge and compass.
1. The drawing shows a compass and straightedge construction of
A. a perpendicular to a given line at a
point on the line
B. the bisector of a given angle
C. an angle congruent to a given angle
D. a perpendicular to a given line from
a point not on the line
Answer: B
2. Use a compass, straightedge, and the drawing below to answer the question.
Which point lies on the line that bisects angle CAB?
A.
B.
C.
D.
T
R
S
Q
Answer: C
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3. Which pair of points determines the perpendicular bisector of line segment AB?
A.
B.
C.
D.
Y, W
Y, Z
X, W
X, Z
Answer: D
G1.1.4 Given a line and a point, construct a line through the point that is parallel to the
original line using straightedge and compass. Given a line and a point, construct a line
through the point that is perpendicular to the original line. Justify the steps of the
constructions.
1. Which point would be on a line perpendicular to l through T?
A.
B.
C.
D.
W
X
Y
Z
Answer: C
2. Use your compass and straightedge to construct a line that is perpendicular to ST and passes
through point O.
Which other point lies on this perpendicular?
A.
B.
C.
D.
W
Z
X
Y
Answer: B
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3. To which point should a line segment from A be drawn so that the resulting figure is a
rectangle?
A.
B.
C.
D.
W
X
Y
Z
Answer: B
G1.1.5 Given a line segment in terms of its endpoints in the coordinate plane, determine its
length and midpoint.
1. The coordinates of the midpoint of AB are
A.
B.
C.
D.
(-2, 5)
(5, 3)
(2, 5)
(-5, 3)
Answer: A
2. Which point is the greatest distance from the origin?
A.
B.
C.
D.
(3, 4)
(9, 2)
(-9, 1)
(-8, -5)
Answer: D
3. The coordinates of the midpoint of AB are (-2, 1), and the coordinates of A are (2, 3). What
are the coordinates of B?
A.
B.
C.
D.
(0, 2)
(-1, 2)
(-6, -1)
(-3, -4)
Answer: C
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G1.1.6 Recognize Euclidean geometry as an axiom system. Know the key axioms and
understand the meaning of and distinguish between undefined terms, axioms, definitions,
and theorems.
1. Plane geometry is based on several undefined terms. Which of the following is an undefined
term?
A.
B.
C.
D.
altitude
median
plane
triangle
Answer: C
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Unit 2
L3.1.1 Convert units of measurement within and between systems; explain how arithmetic
operations on measurements affect units, and carry units through calculations correctly.
1. John needs 180 square feet of tile. The tiles are 3-inch squares and are packed 50 to a box.
What is the minimum number of boxes of tiles John must purchase?
A.
B.
C.
D.
4
15
29
58
Answer: B
G1.2.1 Prove that the angle sum of a triangle is 180° and that an exterior angle of a triangle
is the sum of the two remote interior angles.
1. What is the measure of angle 3?
A.
B.
C.
D.
65°
90°
75°
85°
Answer: D
2. The figure has angle measures as shown.
What is the measure of angle ABD?
A.
B.
C.
D.
150°
70°
80°
30°
Answer: D
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3. In the figure, the measure of angle CAD is twice the measure of angle CAB.
What is the measure of CAB?
A.
B.
C.
D.
60°
120°
45°
30°
Answer: A
G1.2.2 Construct and justify arguments and solve multistep problems involving angle
measure, side length, perimeter, and area of all types of triangles.
1. From shortest to longest, the sides of triangle ABC are
A.
B.
C.
D.
AB, BC, AC
BC, AC, AB
AC, BC, AB
AB, AC, BC
Answer: D
2. In triangle ABC below, AC is 17 units and AB is 8 units.
What is the area, in square units, of triangle ABC?
A.
B.
C.
D.
34
16 3
32 3
68
Answer: A
G1.4.3 Describe and justify hierarchical relationships among quadrilaterals (e.g., every
rectangle is a parallelogram).
1. Which of the following is NOT a property of all parallelograms?
A.
B.
C.
D.
Opposite angles are congruent.
Diagonals form congruent triangles.
Diagonals are perpendicular bisectors.
Consecutive angles are supplementary.
Answer: C
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2. Quadrilateral QRST is placed on a coordinate grid as shown.
What coordinates for S makes QRST a parallelogram?
A.
B.
C.
D.
(8, 6)
(8, 10)
(12, 6)
(12, 10)
Answer: C
G1.4.4 Prove theorems about the interior and exterior angle sums of a quadrilateral.
G1.6.3 Solve problems and justify arguments about central angles, inscribed angles, and
triangles in circles.
1. In circle O, RST formed by chord RS and diameter ST has a measure of 30°. If the
diameter is 12 centimeters, what is the length of chord SR ?
A. 12 3 cm
B. 12 2 cm
C. 6 3 cm
D. 6 2 cm
Answer: C
2. When inscribed in a certain circle, ΔABC intercepts arcs as shown in the diagram. What is
the measure of BAC ?
A.
B.
C.
D.
90°
70°
40°
20°
Answer: D
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3. In the diagram below, A, B, and C are on circle P, and AC  AB .
What is the measure of PCA ?
A.
B.
C.
D.
22.5°
45°
60°
67.5°
Answer: A
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Unit 3
G3.1.1 Define reflection, rotation, translation, and glide reflection and find the image of a
figure under a given isometry.
1. Triangle A’B’C is
A. a reflection of triangle ABC across the y-axis.
B. a 90° clockwise rotation of triangle ABC about
the origin.
C. a translation of triangle ABC across the y-axis.
D. a reflection of triangle ABC across the x-axis.
Answer: A
2. If triangle XYZ is reflected across the y-axis to form triangle X’Y’Z’, what is the coordinate
of Y’?
A.
B.
C.
D.
(-3, 2)
(2, -3)
(3, -2)
(4, 6)
Answer: A
3. The polygon A’B’C’D’E’ is
A.
B.
C.
D.
a reflection of ABCDE across the y-axis.
a reflection of ABCDE across the x-axis.
a translation of ABCDE across the x-axis.
a 180◦ clockwise rotation of ABCDE about the
origin.
Answer: B
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G3.1.2 Given two figures that are images of each other under an isometry, find the
isometry and describe it completely.
1. Triangle A’B’C’ is a transformation of triangle ABC.
If A A’, B B’, and C C’, A’B’C’ is a
A.
B.
C.
D.
reflection of triangle ABC across like l
180° rotation of triangle ABC about point P
translation of triangle ABC across the line l
90° rotation of triangle ABC across the line l
Answer: A
2. On the grid below, triangle A’B’C’ is the result of a two-step transformation of triangle
ABC.
Which best describes the two-step transformation
that was used?
A. a reflection over the x-axis followed by a
translation of 2 units to the right
B. a rotation of 90° clockwise about the origin
followed by a translation of 2 units to the
right
C. a reflection over the x-axis followed by a
reflection over the y-axis
D. a rotation of 180° clockwise about the
origin followed by a translation of 2 units
to the right
Answer: D
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G3.1.3 Find the image of a figure under the composition of two or more isometries and
determine whether the resulting figure is a reflection, rotation, translation, or glide
reflection image of the original figure.
1. Consider this figure.
Which of the following is a rotation in the plane of the given figure?
A.
B.
C.
D.
Answer: A
2. The symmetrical polygon, labeled Position 1, is shown on the grid below.
Which best describes a one-step transformation that
would result in the image of the polygon at Position
2?
A. a translation 5 units down
B. a reflection over the x-axis
C. a 90 counterclockwise rotation about the
origin
D. a 180 clockwise rotation about the point (-5,
0)
Answer: C
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G3.2.1 Know the definition of dilation and find the image of a figure under a given dilation.
1. The rectangle on the grid below will undergo a dilation using (0, 0) as the center and a scale
factor of 2.
Which ordered pair best represents one vertex of the
rectangle that results from this dilation?
A.
B.
C.
D.
(0, 0)
(2, 1)
(2, 4)
(4, 2)
Answer: D
G3.2.2 Given two figures that are images of each other under some dilation, identify the
center and magnitude of the dilation.
1. The large rectangle is a dilation of the small rectangle using a scale factor of 2.
Which best represents the coordinates of the
center point of the dilation?
A.
B.
C.
D.
(0, 0)
(1, 5)
(1, -5)
(-1, -4)
Answer: D
G3.2.3* Find the image of a figure under the composition of a dilation and an isometry.
(Recommended)
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Unit 4
L4.2.1 Know and use the terms of basic logic (e.g., proposition, negation, truth and falsity,
implication, if and only if, contrapositive, and converse).
1. In the proposition below, A and B are both mathematical statements.
A is true if and only if B is true.
Which of the following best defines this proposition?
A.
B.
C.
D.
Both A and B are true.
If B is not true, then A is not true.
B is necessary and sufficient for A.
If A is not true, then B could be true.
Answer: B
L4.2.2 Use the connectives “not,” “and,” “or,” and “if..., then,” in mathematical and
everyday settings. Know the truth table of each connective and how to logically negate
statements involving these connectives.
1. A __________ is a compound statement formed by joining two or more statements with the
word or.
A.
B.
C.
D.
conjunction
negation
truth value
disjunction
Answer: D
2. Consider the conditional statement “If x2 = 36, then x = -6.” All of the following are true
statements except ___________.
A.
B.
C.
D.
(-6)2 = 36
the converse is true.
the statement is true.
the converse is false.
Answer: D
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3. “If I get a B on the test, I will pass.” What is the underlined portion called in this conditional
statement?
A.
B.
C.
D.
the conclusion
the hypothesis
the argument
the converse
Answer: B
L4.2.3 Use the quantifiers “there exists” and “all” in mathematical and everyday settings
and know how to logically negate statements involving them.
1. According to the Venn diagram, which is true?
A.
B.
C.
D.
No football players play offense and defense.
All football players play offense or defense.
Some football players play offense and defense.
All football players play defense.
Answer: C
2. Which of the following can be used to show that the statement below is false?
For every polygon, there exists a polygon with one less side.
A.
B.
C.
D.
a converse
a conjunction
a contrapositive
a counterexample
Answer: D
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L4.2.4 Write the converse, inverse, and contrapositive of an “If..., then...” statement. Use
the fact, in mathematical and everyday settings, that the contrapositive is logically
equivalent to the original while the inverse and converse are not.
1. Write the symbolic statement in conditional or biconditional form and determine whether it is
true or false. ~p  ~q
p = two angles are supplementary
q = the sum of their measure is 180°
A. If two angles are supplementary, then the sum of their measures is 180°. True.
B. If two angles are not supplementary, then the sum of their measures is not 180°.
False.
C. If two angles are not supplementary, then the sum of their measures is 180°. False.
D. If two angles are not supplementary, then the sum of their measures is not 180°. True.
Answer: D
2. What is the contrapositive of the statement below?
If a triangle is isosceles, then it has two congruent sides.
A.
B.
C.
D.
If a triangle does not have two congruent sides, then it is not isosceles.
If a triangle is isosceles, then it does not have two congruent sides.
If a triangle has two congruent sides, then it is isosceles.
If a triangle is not isosceles, then it does not have two congruent sides.
Answer: A
3. Consider the theorem stated below.
If a quadrilateral is a rhombus, then the diagonals are perpendicular.
Which of the following is the inverse of this theorem?
A.
B.
C.
D.
If a quadrilateral is not a rhombus, then the diagonals are not perpendicular.
If the diagonals of a quadrilateral are perpendicular, then it is a rhombus.
If a quadrilateral is a rhombus, then the diagonals are not perpendicular.
If the diagonals of a quadrilateral are not perpendicular, then it is not a rhombus.
Answer: A
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L4.3.1 Know the basic structure for the proof of an “If..., then...” statement (assuming the
hypothesis and ending with the conclusion) and that proving the contrapositive is
equivalent.
1. Makito wants to give a direct proof for the theorem below.
If two angles are vertical angles, then they are congruent.
First he wants to draw a diagram that would be most useful for his proof. Which of the
following angles should Makito draw?
A.
B.
C.
D.
right angles
vertical angles
adjacent angles
congruent angles
Answer: B
L4.3.2 Construct proofs by contradiction. Use counterexamples, when appropriate, to
disprove a statement.
1. One possible counterexample for the statement “All fruits taste sweet” is ________.
A.
B.
C.
D.
Lemons
Oranges
Plums
Watermelons
Answer: A
2. Milo plans to prove that no isosceles right triangle has whole-unit measurements for all three
sides. To construct a proof by contradiction, which should be Milo’s first step?
A.
B.
C.
D.
Show that a2 + b2 = c2.
Assume that such a triangle exists.
Pick three whole numbers that form a Pythagorean Triple.
Assume no triangle has whole-unit measurements for all three sides.
Answer: B
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G1.1.2 Solve multistep problems and construct proofs involving corresponding angles,
alternate interior angles, alternate exterior angles, and same-side (consecutive) interior
angles.
1. The figure shows line l intersecting lines r and s.
In the figure, angle 1 and angle 2 are ________
A.
B.
C.
D.
corresponding angles
alternate interior angles
consecutive angles
alternate exterior angles
Answer: A
2. In the figure, lines l and m are cut by the transversal t forming the angles shown.
Angle 3 and angle 6 are ____
A.
B.
C.
D.
Alternate interior angles
Alternate exterior angles
Vertical angles
Corresponding angles
Answer: A
3. Line l intersects lines w, x, y, and z. Which two lines are parallel?
A.
B.
C.
D.
Line y and line z
Line w and line y
Line x and line z
Line w and line x
Answer: C
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G1.2.5 Solve multistep problems and construct proofs about the properties of medians,
altitudes, and perpendicular bisectors to the sides of a triangle, and the angle bisectors of a
triangle. Using a straightedge and compass, construct these lines.
1. In the triangle below, angle D is a right angle.
Which point best represents the intersection of the perpendicular bisectors of the sides of the
triangle?
A.
B.
C.
D.
Point A
Point B
Point C
Point D
Answer: A
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Unit 5
L2.1.6 Recognize when exact answers aren’t always possible or practical. Use appropriate
algorithms to approximate solutions to equations (e.g., to approximate square roots).
1. The unit lengths of the legs of a right triangle are shown in the diagram.
What is the value of x to the nearest whole unit?
A.
B.
C.
D.
10
11
12
13
Answer: B
G1.2.3 Know a proof of the Pythagorean Theorem and use the Pythagorean Theorem and
its converse to solve multistep problems.
1. Triangle ABC is a right triangle with the measures shown.
The length of BC is ______
A.
B.
C.
D.
24 in.
32 in.
576 in.
18 in.
Answer: A
2. A customer provided this diagram of a patio to a fencing company.
What is the length of the unlabeled side?
A.
B.
C.
D.
12 ft
13 ft
10 ft
11 ft
Answer: C
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3. The top of a ladder is leaning on a building at a point 12 feet above the ground; the bottom of
the ladder is 5 feet from the base of the building. What is the length of the ladder?
A.
B.
C.
D.
19 ft
7 ft
13 ft
17 ft
Answer: C
G1.2.4 Prove and use the relationships among the side lengths and the angles of 30º- 60º90º triangles and 45º- 45º- 90º triangles.
1. The drawing shows the measurements in a section of a circular design. How long is the
radius of the circle?
A.
B.
C.
D.
8.7 cm
4.3 cm
10 cm
7cm
Answer: C
2. A carpenter is building a flight of stairs as pictured in the drawing.
What is the horizontal distance from the foot of the stair to the wall?
A.
B.
C.
D.
14.1 ft
17.3 ft
20.0 ft
28.3 ft
Answer: B
3. A design is formed by joining isosceles right triangles and 60°-30° right triangles as shown in
the diagram. If the hypotenuse of the 60°-30° triangle is 12 centimeters, which is closest to
the length of one leg of the isosceles right triangle?
A.
B.
C.
D.
7.2 cm
8.5 cm
10.4 cm
6 cm
Answer: D
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G2.1.1 Know and demonstrate the relationships between the area formula of a triangle, the
area formula of a parallelogram, and the area formula of a trapezoid.
1. In parallelogram ABCD below, the length of AC is j units, and the length of BE is k units.
Which expression represents the area, in square units, of ABCD?
1
A.
jk
2
B. jk
1
C. ( jk)2
2
D. (jk)2
Answer: B
2. The parallelogram has the measurements shown.
Which is the closest to the length of the altitude, h?
A.
B.
C.
D.
19.63
8.91
8.67
6.81
Answer: C
G2.1.2 Know and demonstrate the relationships between the area formulas of various
quadrilaterals (e.g., explain how to find the area of a trapezoid based on the areas of
parallelograms and triangles).
1. The lengths of the sides of trapezoid ABCD are shown below.
Which expression shows the square-unit area of
the trapezoid as the sum of the areas of the
rectangle and a triangle?
A. (4 + 8) + (4 2 + 12)
1
B. (4  8) + (4  4)
2
1
C.
(8 + 12) + (4  4 2 )
2
1
1
D.
(8 + 12) + (4  4 2 )
2
2
Answer: B
Geometry Assessment – August 2008 Revision
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G1.5.1 Know and use subdivision or circumscription methods to find areas of polygons
(e.g., regular octagon, nonregular pentagon).
1. The coordinates of the vertices shown in the drawing below correspond to points in the
coordinate plane.
What is the area, in square units, of the figure?
A.
B.
C.
D.
86
224
344
576
Answer: B
G1.5.2 Know, justify, and use formulas for the perimeter and area of a regular n-gon and
formulas to find interior and exterior angles of a regular n-gon and their sums.
1. Which expression is equivalent to the perimeter of a regular n-gon with sides 17 units in
length?
A. 17n
B. n + 17
C. 172
17
D.
n
Answer: A
2. What are the values of x and y?
A.
B.
C.
D.
x = 91°, y = 98°
x = 91°, y = 108°
x = 101°, y = 98°
x = 10°1, y = 108°
Answer: A
Geometry Assessment – August 2008 Revision
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3. The two adjacent figures are a regular hexagon and a regular octagon. What is the measure of
 PQR?
A.
B.
C.
D.
87.5°
90°
105°
120°
Answer: C
L1.1.6 Explain…the importance of π because of its role in circle relationships…
1. Which of the following radii yields a circle with an area that can be expressed as a rational
number?
A. radius =  inches
1
B. radius =
inch

1
C. radius =
inch

1
D. radius =
inch
2
Answer: C
G1.6.1 Solve multistep problems involving circumference and area of circles.
2. A square is inscribed in a circle with a diameter of 10 units, as shown below.
What is the area, in square units, of the shaded region?
A.
B.
C.
D.
25  - 50
25  - 100
100  - 50
100  - 100
Answer: A
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3. This is a sketch of a stained-glass window in the shape of a semicircle.
Ignoring the seams, how much glass is needed for the window?
A.
B.
C.
D.
4  sq ft
8  sq ft
12  sq ft
16  sq ft
Answer: B
G1.6.4 Know and use properties of arcs and sectors and find lengths of arcs and areas of
sectors.
1. A circle for a game spinner is divided into 3 regions as shown. RP is a diameter. What is the
area of the shaded sector ROS if RP = 8?
A.
B.
C.
D.
1.5 
6
24 
72 
Answer: B
2. The circle shown below has its center at the origin with point A(6, 0) and point B on the
circle.
What is the length of AB ?
A.
B.
C.
D.
1 unit
2  units
4 units
6 units
Answer: D
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L1.1.6 Explain the importance of the irrational numbers √2 and √3 in basic right triangle
trigonometry…
1. If a=3 3 in the right triangle below, what is the value of b?
A.
B.
C.
D.
9
6 3
12 3
18
Answer: A
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Unit 6
G2.3.5 Know and apply the theorem stating that the effect of a scale factor of k relating one
two-dimensional figure to another or one three-dimensional figure to another, on the
length, area, and volume of the figures is to multiply each by k, k2, and k3, respectively.
(target: 2-D part)
1. The radius of a circular pizza pan for a large pizza is double the radius for a medium pizza.
Assuming both pans are covered with the same thickness of pizza, what is the amount of
pizza in a large size compared to a medium size?
A.
B.
C.
D.
twice the amount
four times the amount
six times the amount
 2 times the amount
Answer: B
G1.8.1 Solve multistep problems involving surface area and volume of pyramids, prisms,
cones, cylinders, hemispheres, and spheres.
1. What is the volume, in terms of  , of the right circular cylinder below?
A.
B.
C.
D.
100 
150 
250 
300 
cubic units
cubic units
cubic units
cubic units
Answer: C
2. What is the approximate volume of a can that is 5 inches tall and has a 2.5 inch diameter?
A.
B.
C.
D.
19.6 cu in.
24.5 cu in.
39.3 cu in.
98.1 cu in.
Answer: B
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G1.8.2 Identify symmetries of pyramids, prisms, cones, cylinders, hemispheres, and
spheres.
1. A plane is parallel to and equidistant from the bases of a right circular cylinder. What is the
shape of the resulting planar cross-section?
A.
B.
C.
D.
circle
ellipse
hemisphere
sphere
Answer: A
G2.1.3 Know and use the relationship between the volumes of pyramids and prisms (of
equal base and height) and cones and cylinders (of equal base and height).
1. What is the ratio of the volume of a square pyramid to the volume of a cube if they have the
same height and congruent bases?
A.
B.
C.
D.
1:2
1:3
2:1
3:1
Answer: B
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G2.2.1 Know and use the relationship between the volumes of pyramids and prisms (of
equal base and height) and cones and cylinders (of equal base and height).
1. The net shown below will be folded along some of the grid lines, without any overlapping, to
create a three-dimensional figure.
Which best represents the resulting three-dimensional figure?
A.
B.
C.
D.
Answer: A
G2.2.2 Identify or sketch cross sections of three-dimensional figures. Identify or sketch
solids formed by revolving two-dimensional figures around lines.
1. A plane that cuts a rectangular prism contains one top edge and the opposite bottom edge of
the prism. What three-dimensional figures result from this planar cut?
A.
B.
C.
D.
right circular cones
square pyramids
Triangular pyramids
triangular prisms
Answer: D
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G2.3.5 Know and apply the theorem stating that the effect of a scale factor of k relating one
two-dimensional figure to another or one three-dimensional figure to another, on the
length, area, and volume of the figures is to multiply each by k, k2, and k3, respectively.
(target: 3-D part)
1. This is a scale drawing of a building. What is the actual height of the building?
A.
B.
C.
D.
58.5 m
71.5 m
78 m
84.5 m
Answer: A
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Unit 7
G2.3.1Prove that triangles are congruent using the SSS, SAS, ASA, and AAS criteria and
that right triangles are congruent using the hypotenuse-leg criterion.
1. A partial proof for proving two triangles congruent is shown below.
Given: ABCD is a rectangle
Prove: ΔABC  ΔCDA
Which statement should be used in Step 3 of this proof?
AD  AB
A.
B.
C.
D.
ADC  CBA
ADC and CBA are right triangles
AC  AC
Answer: D
2. Use the proof to answer the question below.
What reason can be used to prove that the triangles are congruent?
A.
B.
C.
D.
AAS
ASA
SAS
SSS
Answer: D
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G2.3.2 Use theorems about congruent triangles to prove additional theorems and solve
problems, with and without use of coordinates.
1. A median to the base of an isosceles triangle is constructed, and the resulting two triangles
are shown to be congruent by SAS. Which of the following statements is a corollary to this
proof?
A.
B.
C.
D.
The vertex of an isosceles triangle cannot be a right angle.
The side opposite the vertex angle is the longest side of the triangle.
The angles opposite congruent sides of an isosceles triangle are congruent.
The vertex angle of an isosceles must be greater than the other angles.
Answer: C
2. Which of the following facts would be sufficient to prove that triangles ABC and DBE are
similar?
A.
B.
C.
D.
CE and BE are congruent.
ACE is a right triangle.
AC and DE are parallel.
A and B are congruent.
Answer: C
G2.3.3 Prove that triangles are similar by using SSS, SAS, and AA conditions for
similarity.
1. In the figure below, AC  DF and A  D .
Which additional information would be enough to
prove that ΔABC  ΔDEF?
A.
B.
C.
D.
AB  DE
AB  BC
BC  EF
BC  DE
Answer: A
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2. Howard drew the two right triangles below. He did not show measurements but stated the
two triangles were similar.
Which postulate or theorem can Howard use to prove
these triangles are similar?
A.
B.
C.
D.
AA
ASA
HL
SAS
Answer: A
G2.3.4 Use theorems about similar triangles to solve problems with and without use of
coordinates.
1. Given: AB and CD intersect at point E; 1  2
Which theorem or postulate can be used to prove ΔAED~ΔBEC?
A.
B.
C.
D.
AA
SSS
ASA
SAS
Answer: A
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2. Which of the following is similar to the triangle shown at the right?
A.
B.
C.
D.
Answer: A
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Unit 8
L1.2.3 Use vectors to represent quantities that have magnitude and direction, interpret
direction and magnitude of a vector numerically, and calculate the sum and difference of
two vectors.
1. An object is acted upon simultaneously by a 100-pound force due north and a 100-pound
force due east, as represented by the diagram below.
What is the resultant force and its direction?
A.
B.
C.
D.
200 2 pounds NE
200 pounds NE
100 2 pounds NE
100 pounds NE
Answer: C
2. If
 12 
A.   
 7
  7
B.  
 12 
 12
C.   
 1
  1
D.  
 12
Answer: C
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3. An airplane is headed due north at 300 nautical miles per hour (knots) as shown in the
drawing. The wind is directly from the east at 50 knots.
Which is closest to the resultant speed of the airplane?
A.
B.
C.
D.
314 knots
304 knots
296 knots
286 knots
Answer: B
G1.3.1 Define the sine, cosine, and tangent of acute angles in a right triangle as ratios of
sides. Solve problems about angles, side lengths, or areas using trigonometric ratios in right
triangles.
1. Approximately how many feet tall is the streetlight?
A.
B.
C.
D.
12.8
15.4
16.8
23.8
Answer: C
2. In the accompanying diagram, m A = 32° and AC = 10. Which equation could be used to
find x in ΔABC?
A. x = 10sin 32°
B. x = 10cos 32°
C. x = 10tan 32°
10
D. x =
cos 32
Answer: C
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3. In the figure below, if sin x =
5
, what are cos x and tan x?
13
A. cos x =
12
5
and tan x =
13
12
B. cos x =
12
12
and tan x =
13
5
C. cos x =
13
5
and tan x =
12
12
D. cos x =
13
13
and tan x =
12
5
Answer: A
4. A cable 48 feet long stretches from the top of a pole to the ground. If the cable forms a 40°
angle with the ground, which is closest to the height of the pole?
sin 40°  0.642
cos 40°  0.766
tan 40°  0.839
A.
B.
C.
D.
26.4 ft
30.9 ft
36.8 ft
40.3 ft
Answer: B
G1.3.2 Know and use the Law of Sines and the Law of Cosines and use them to solve
problems. Find the area of a triangle with sides a and b and included angle θ using the
formula Area = (1/2) a b sin θ.
G1.3.3 Determine the exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, and their
integer multiples and apply in various contexts.
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Unit 9
G1.4.1 Solve multistep problems and construct proofs involving angle measure, side length,
diagonal length, perimeter, and area of squares, rectangles, parallelograms, kites, and
trapezoids.
1. A rectangle with vertices of (5, 5), (5, -5), (-5, 5), (-5, -5) is shown below.
What is the length of one of the diagonals of the rectangle?
A.
B.
C.
D.
2 5
10
10 2
20
Answer: C
2. In parallelogram ABCD, what is the measure of  ACD?
A.
B.
C.
D.
70°
45°
35°
25°
Answer: B
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3. ABCD is a rhombus.
What is the measure of  CBD?
A.
B.
C.
D.
50°
60°
70°
75°
Answer: C
G1.4.2 Solve multistep problems and construct proofs involving quadrilaterals (e.g., prove
that the diagonals of a rhombus are perpendicular) using Euclidean methods or coordinate
geometry.
1. To determine the sum of the measures of the interior angles of a quadrilateral, Kim used the
five steps shown below.
1. Identify a point, P, on the interior of the quadrilateral.
2. Draw lines from P to each of the vertices, forming 4
triangles.
3. State that for each of the triangles, the sum of the measures
of the angles is 180°.
4. 4  180 = 720
5. Therefore, the sum of the measures of the angles is 720°.
Which statement below explains the error that Kim made?
A.
B.
C.
D.
Kim included the measures of the angles at point P, so she needed to subtract 360°.
Kim could not begin her proof by inserting a point P inside the quadrilateral.
Kim needed to first find the sum of the measures of the exterior angles.
Kim should have picked a point P outside the quadrilateral.
Answer: A
2. In rectangle ABCD, which of the following pairs of segments are not necessarily congruent?
A.
B.
C.
D.
BD and AC
AB and CD
BC and DC
BE and CE
Answer: C
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3. Quadrilateral ABCD is a parallelogram
Which of the following must be true?
AB  AD
AC  BD
A  D
B  D
A.
B.
C.
D.
Answer: D
G1.4.5* Understand the definition of a cyclic quadrilateral and know and use the basic
properties of cyclic quadrilaterals. (Recommended)
G1.6.2 Solve problems and justify arguments about chords (e.g., if a line through the center
of a circle is perpendicular to a chord, it bisects the chord) and lines tangent to circles (e.g.,
a line tangent to a circle is perpendicular to the radius drawn to the point of tangency).
1. Which angle pair is formed by a tangent of a circle and the diameter of the circle at the point
of tangency?
A.
B.
C.
D.
complementary
nonadjacent
vertical
right
Answer: D
2. Chords AB and CD intersect at R. Using the values shown in the diagram, what is the
measure of RB ?
A.
B.
C.
D.
6
7.5
8
9.5
Answer: B
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3. RB is tangent to a circle, whose center is A, at point B, BD is a diameter.
What is m  CBR?
A. 50°
B. 65°
C. 90°
D. 130°
Answer: B
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