Download doc - GLCE/HSCE: Geometry Assessment · Web viewThe polygon A’B’C’D’E’ is a reflection of ABCDE across the y-axis. a reflection of ABCDE across the x-axis. a translation of ABCDE

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. conjecturingB. inductive reasoningC. deductive reasoningD. 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. Some musicians like music.B. Some happy people do not like music.C. All musicians like music.D. All happy people are musicians.

Answer: A

Geometry Assessment – August 2008 Revision 1

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. More customers buy roses than carnations.B. The salesperson likes carnations.C. Jasmine will be excited to receive flowers for her birthday.D. 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. theoremB. proofC. definitionD. postulate

Answer: D


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. If two planes intersect, then their intersection is a line.B. If two lines intersect, then their intersection is exactly one point.C. Through any three points not on the same line, there is exactly one plane.D. 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. An angle measures 56° and is acute. B. All acute angles have measures between 0° and 90°.C. An angle measures 48° if and only if it is acute. D. 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. AB = ADB. angle A and angle B are supplementaryC. measure of angle B = measure of angle DD. AC = BD

Answer: C


2. Which is a necessary and sufficient condition for a parallelogram to be classified as a rectangle?

A. Opposite sides are congruent.B. Opposite sides are parallel.C. Diagonals bisect each other.D. 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. 70B. 85C. 80D. 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. 80°B. 130°C. 50°D. 100°

Answer: C


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. 55°B. 125°C. 35°D. 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 angleC. an angle congruent to a given angleD. 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. TB. RC. SD. Q

Answer: C


3. Which pair of points determines the perpendicular bisector of line segment AB?

A. Y, WB. Y, ZC. X, WD. 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. WB. XC. YD. 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. WB. ZC. XD. Y

Answer: B


3. To which point should a line segment from A be drawn so that the resulting figure is a rectangle?

A. WB. XC. YD. 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. (-2, 5)B. (5, 3)C. (2, 5)D. (-5, 3)

Answer: A

2. Which point is the greatest distance from the origin?

A. (3, 4)B. (9, 2)C. (-9, 1)D. (-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. (0, 2)B. (-1, 2)C. (-6, -1)D. (-3, -4)

Answer: C


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. altitudeB. medianC. planeD. triangle

Answer: C


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. 4B. 15C. 29D. 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. 65°B. 90°C. 75°D. 85°

Answer: D 2. The figure has angle measures as shown.

What is the measure of angle ABD?

A. 150°B. 70°C. 80°D. 30°

Answer: D


3. In the figure, the measure of angle CAD is twice the measure of angle CAB.

What is the measure of CAB?

A. 60°B. 120°C. 45°D. 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. AB, BC, ACB. BC, AC, ABC. AC, BC, ABD. 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. 34B. 16C. 32D. 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. Opposite angles are congruent.B. Diagonals form congruent triangles.C. Diagonals are perpendicular bisectors.D. Consecutive angles are supplementary.

Answer: C


2. Quadrilateral QRST is placed on a coordinate grid as shown.

What coordinates for S makes QRST a parallelogram?

A. (8, 6)B. (8, 10)C. (12, 6)D. (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, formed by chord and diameter has a measure of 30°. If the diameter is 12 centimeters, what is the length of chord ?

A. 12 cmB. 12 cmC. 6 cmD. 6 cm

Answer: C

2. When inscribed in a certain circle, ΔABC intercepts arcs as shown in the diagram. What is the measure of ?

A. 90°B. 70°C. 40°D. 20°

Answer: D


3. In the diagram below, A, B, and C are on circle P, and .

What is the measure of ?

A. 22.5°B. 45°C. 60°D. 67.5°

Answer: A


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. (-3, 2)B. (2, -3)C. (3, -2)D. (4, 6)

Answer: A

3. The polygon A’B’C’D’E’ is

A. a reflection of ABCDE across the y-axis.B. a reflection of ABCDE across the x-axis.C. a translation of ABCDE across the x-axis.D. a 180◦ clockwise rotation of ABCDE about the

origin.

Answer: B


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 Aà A’, Bà B’, and Cà C’, A’B’C’ is a

A. reflection of triangle ABC across like lB. 180° rotation of triangle ABC about point PC. translation of triangle ABC across the line lD. 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


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 downB. a reflection over the x-axisC. a 90 counterclockwise rotation about the

originD. a 180 clockwise rotation about the point (-5,

0)

Answer: C


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. (0, 0)B. (2, 1)C. (2, 4)D. (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. (0, 0)B. (1, 5)C. (1, -5)D. (-1, -4)

Answer: D

G3.2.3* Find the image of a figure under the composition of a dilation and an isometry. (Recommended)


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. Both A and B are true.B. If B is not true, then A is not true.C. B is necessary and sufficient for A.D. 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. conjunctionB. negationC. truth valueD. disjunction

Answer: D

2. Consider the conditional statement “If x2 = 36, then x = -6.” All of the following are true statements except ___________.

A. (-6)2 = 36B. the converse is true.C. the statement is true.D. the converse is false.

Answer: D


3. “If I get a B on the test, I will pass.” What is the underlined portion called in this conditional statement?

A. the conclusionB. the hypothesisC. the argumentD. 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. No football players play offense and defense.B. All football players play offense or defense.C. Some football players play offense and defense.D. 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. a converseB. a conjunctionC. a contrapositive D. a counterexample

Answer: D


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 à ~q

p = two angles are supplementaryq = 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. If a triangle does not have two congruent sides, then it is not isosceles. B. If a triangle is isosceles, then it does not have two congruent sides. C. If a triangle has two congruent sides, then it is isosceles. D. 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. If a quadrilateral is not a rhombus, then the diagonals are not perpendicular.B. If the diagonals of a quadrilateral are perpendicular, then it is a rhombus.C. If a quadrilateral is a rhombus, then the diagonals are not perpendicular.D. If the diagonals of a quadrilateral are not perpendicular, then it is not a rhombus.

Answer: A


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. right anglesB. vertical anglesC. adjacent anglesD. 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. LemonsB. OrangesC. PlumsD. 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. Show that a2 + b2 = c2.B. Assume that such a triangle exists.C. Pick three whole numbers that form a Pythagorean Triple.D. Assume no triangle has whole-unit measurements for all three sides.

Answer: B


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. corresponding anglesB. alternate interior anglesC. consecutive anglesD. 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. Alternate interior anglesB. Alternate exterior anglesC. Vertical anglesD. Corresponding angles

Answer: A

3. Line l intersects lines w, x, y, and z. Which two lines are parallel?

A. Line y and line zB. Line w and line yC. Line x and line zD. Line w and line x

Answer: C


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. Point AB. Point BC. Point CD. Point D

Answer: A


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. 10B. 11C. 12D. 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. 24 in.B. 32 in.C. 576 in.D. 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. 12 ftB. 13 ftC. 10 ftD. 11 ft

Answer: C


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. 19 ftB. 7 ftC. 13 ftD. 17 ft

Answer: C

G1.2.4 Prove and use the relationships among the side lengths and the angles of 30º- 60º- 90º 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. 8.7 cmB. 4.3 cmC. 10 cmD. 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. 14.1 ftB. 17.3 ftC. 20.0 ftD. 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. 7.2 cmB. 8.5 cmC. 10.4 cmD. 6 cm

Answer: D


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 is j units, and the length of is k units.

Which expression represents the area, in square units, of ABCD?

A. jk

B. jk

C. ( jk)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. 19.63B. 8.91C. 8.67D. 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 + 12)

B. (4 8) + (4 4)

C. (8 + 12) + (4 4 )

D. (8 + 12) + (4 4 )

Answer: BG1.5.1 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. 86B. 224C. 344D. 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. 17nB. n + 17C. 172

D.

Answer: A

2. What are the values of x and y?

A. x = 91°, y = 98°B. x = 91°, y = 108°C. x = 101°, y = 98°D. x = 10°1, y = 108°

Answer: A


3. The two adjacent figures are a regular hexagon and a regular octagon. What is the measure of PQR?

A. 87.5°B. 90°C. 105°D. 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

B. radius = inch

C. radius = inch

D. radius = inch

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. 25 - 50B. 25 - 100C. 100 - 50D. 100 - 100

Answer: A


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. 4 sq ftB. 8 sq ftC. 12 sq ftD. 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. is a diameter. What is the area of the shaded sector ROS if RP = 8?

A. 1.5B. 6C. 24D. 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 ?

A. 1 unitB. 2 unitsC. 4 unitsD. 6 units

Answer: D


L1.1.6 Explain the importance of the irrational numbers √2 and √3 in basic right triangle trigonometry…

1. If a=3 in the right triangle below, what is the value of b?

A. 9B. 6C. 12D. 18

Answer: A


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. twice the amountB. four times the amountC. six times the amountD. 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. 100 cubic unitsB. 150 cubic unitsC. 250 cubic unitsD. 300 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. 19.6 cu in.B. 24.5 cu in.C. 39.3 cu in.D. 98.1 cu in.

Answer: B


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. circleB. ellipseC. hemisphereD. 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. 1:2B. 1:3C. 2:1D. 3:1

Answer: B


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. right circular conesB. square pyramidsC. Triangular pyramidsD. triangular prisms

Answer: D


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. 58.5 mB. 71.5 mC. 78 mD. 84.5 m

Answer: A


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 rectangleProve: ΔABC ΔCDA

Which statement should be used in Step 3 of this proof?

A. B. C. and are right trianglesD.

Answer: D

2. Use the proof to answer the question below.

What reason can be used to prove that the triangles are congruent?

A. AASB. ASAC. SASD. SSS

Answer: D


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. The vertex of an isosceles triangle cannot be a right angle.B. The side opposite the vertex angle is the longest side of the triangle.C. The angles opposite congruent sides of an isosceles triangle are congruent. D. 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. and are congruent.B. is a right triangle.C. and are parallel.D. and 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, and .

Which additional information would be enough to prove that ΔABC ΔDEF?

A. B. C. D.

Answer: A


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. AAB. ASAC. HLD. SAS

Answer: A

G2.3.4 Use theorems about similar triangles to solve problems with and without use of coordinates.

1. Given: and intersect at point E;

Which theorem or postulate can be used to prove ΔAED~ΔBEC?

A. AAB. SSSC. ASAD. SAS

Answer: A


2. Which of the following is similar to the triangle shown at the right?

A.

B.

C.

D.

Answer: A


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. 200 pounds NEB. 200 pounds NEC. 100 pounds NED. 100 pounds NE

Answer: C

2. If

A.

B.

C.

D.

Answer: C


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. 314 knotsB. 304 knotsC. 296 knotsD. 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. 12.8B. 15.4C. 16.8D. 23.8

Answer: C

2. In the accompanying diagram, m = 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°

D. x =

Answer: C


3. In the figure below, if sin x = , what are cos x and tan x?

A. cos x = and tan x =

B. cos x = and tan x =

C. cos x = and tan x =

D. cos x = and tan x =

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° 0.642cos 40° 0.766tan 40° 0.839

A. 26.4 ftB. 30.9 ftC. 36.8 ftD. 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.


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. 2B. 10C. 10D. 20

Answer: C

2. In parallelogram ABCD, what is the measure of ACD?

A. 70°B. 45°C. 35°D. 25°

Answer: B


3. ABCD is a rhombus.

What is the measure of CBD?

A. 50°B. 60°C. 70°D. 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 = 7205. Therefore, the sum of the measures of the angles is 720°.

Which statement below explains the error that Kim made?

A. Kim included the measures of the angles at point P, so she needed to subtract 360°. B. Kim could not begin her proof by inserting a point P inside the quadrilateral. C. Kim needed to first find the sum of the measures of the exterior angles.D. 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. and B. and C. and D. and

Answer: C


3. Quadrilateral ABCD is a parallelogram

Which of the following must be true?

A. B. C. A DD. B 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. complementaryB. nonadjacentC. verticalD. right

Answer: D

2. Chords and intersect at R. Using the values shown in the diagram, what is the measure of ?

A. 6B. 7.5C. 8D. 9.5

Answer: B


3. is tangent to a circle, whose center is A, at point B, is a diameter.

What is m CBR?A. 50°B. 65°C. 90°D. 130°

Answer: B
