keystone exams: geometry

Keystone Exams: GeometryAssessment Anchors and Eligible Content

with Sample Questions and Glossary

Pennsylvania Department of Education

www.education.state.pa.us

April 2014

Pennsylvania Department of Education—Assessment Anchors and Eligible Content

PENNSYLVANIA DEPARTMENT OF EDUCATION

General Introduction to the Keystone Exam Assessment Anchors

Introduction

Since the introduction of the Keystone Exams, the Pennsylvania Department of Education (PDE) has been working to create a set of tools designed to help educators improve instructional practices and better understand the Keystone Exams. The Assessment Anchors, as defined by the Eligible Content, are one of the many tools the Department believes will better align curriculum, instruction, and assessment practices throughout the Commonwealth. Without this alignment, it will not be possible to significantly improve student achievement across the Commonwealth.

How were Keystone Exam Assessment Anchors developed?

Prior to the development of the Assessment Anchors, multiple groups of PA educators convened to create a set of standards for each of the Keystone Exams. Enhanced Standards, derived from a review of existing standards, focused on what students need to know and be able to do in order to be college and career ready. (Note: Since that time, PA Core Standards have replaced the Enhanced Standards and reflect the college- and career-ready focus.) Additionally, the Assessment Anchors and Eligible Content statements were created by other groups of educators charged with the task of clarifying the standards assessed on the Keystone Exams. The Assessment Anchors, as defined by the Eligible Content, have been designed to hold together, or anchor, the state assessment system and the curriculum/instructional practices in schools.

Assessment Anchors, as defined by the Eligible Content, were created with the following design parameters: Clear: The Assessment Anchors are easy to read and are user friendly; they clearly detail which

standards are assessed on the Keystone Exams.

Focused: The Assessment Anchors identify a core set of standards that can be reasonably assessed on a large-scale assessment; this will keep educators from having to guess which standards are critical.

Rigorous: The Assessment Anchors support the rigor of the state standards by assessing higher-order and reasoning skills.

Manageable: The Assessment Anchors define the standards in a way that can be easily incorporated into a course to prepare students for success.

How can teachers, administrators, schools, and districts use these Assessment Anchors?

The Assessment Anchors, as defined by the Eligible Content, can help focus teaching and learning because they are clear, manageable, and closely aligned with the Keystone Exams. Teachers and administrators will be better informed about which standards will be assessed. The Assessment Anchors and Eligible Content should be used along with the Standards and the Curriculum Framework of the Standards Aligned System (SAS) to build curriculum, design lessons, and support student achievement.

The Assessment Anchors and Eligible Content are designed to enable educators to determine when they feel students are prepared to be successful in the Keystone Exams. An evaluation of current course offerings, through the lens of what is assessed on those particular Keystone Exams, may provide an opportunity for an alignment to ensure student preparedness.


How are the Assessment Anchors organized?

The Assessment Anchors, as defined by the Eligible Content, are organized into cohesive blueprints, each structured with a common labeling system that can be read like an outline. This framework is organized first by module, then by Assessment Anchor, followed by Anchor Descriptor, and then finally, at the greatest level of detail, by an Eligible Content statement. The common format of this outline is followed across the Keystone Exams.

Here is a description of each level in the labeling system for the Keystone Exams: Module: The Assessment Anchors are organized into two thematic modules for each of the

Keystone Exams. The module title appears at the top of each page. The module level is important because the Keystone Exams are built using a module format, with each of the Keystone Exams divided into two equal-size test modules. Each module is made up of two or more Assessment Anchors.

Assessment Anchor: The Assessment Anchor appears in the shaded bar across the top of each Assessment Anchor table. The Assessment Anchors represent categories of subject matter that anchor the content of the Keystone Exams. Each Assessment Anchor is part of a module and has one or more Anchor Descriptors unified under it.

Anchor Descriptor: Below each Assessment Anchor is a specific Anchor Descriptor. The Anchor Descriptor level provides further details that delineate the scope of content covered by the Assessment Anchor. Each Anchor Descriptor is part of an Assessment Anchor and has one or more Eligible Content statements unified under it.

Eligible Content: The column to the right of the Anchor Descriptor contains the Eligible Content statements. The Eligible Content is the most specific description of the content that is assessed on the Keystone Exams. This level is considered the assessment limit and helps educators identify the range of the content covered on the Keystone Exams.

PA Core Standard: In the column to the right of each Eligible Content statement is a code representing one or more PA Core Standards that correlate to the Eligible Content statement. Some Eligible Content statements include annotations that indicate certain clarifications about the scope of an Eligible Content.

“e.g.” (“for example”)—sample approach, but not a limit to the Eligible Content

“Note”—content exclusions or definable range of the Eligible Content

How do the K–12 Pennsylvania Core Standards affect this document?

Assessment Anchor and Eligible Content statements are aligned to the PA Core Standards; thus, the former enhanced standards are no longer necessary. Within this document, all standard references reflect the PA Core Standards.

Standards Aligned System—www.pdesas.org

Pennsylvania Department of Education—www.education.state.pa.us


Keystone Exams: Geometry

Properties of Circles

x°

n°Tangent-Chord

x = n12

a

cx°

d

b

m° n°

2 Chords

a · b = c · d

x = (m + n)12

a

c

x°b

m°n°

Tangent-Secant

a2 = b (b + c)

x = (m − n)12

a

x°

b

m° n°

2 Tangents

a = b

x = (m − n)12

Inscribed Angle

x = n12

x°n°

2 Secants

b (a + b) = d (c + d )

x = (m − n)12

a

cx°

d

bm° n°

Angle measure is represented by x. Arc measure is represented by m and n. Lengths are given by a, b, c, and d.

Right Triangle Formulas

a

b

c

Trigonometric Ratios:

opposite

adjacent

hypotenuse

θ

opposite

hypotenusesin θ =

adjacent

hypotenusecos θ =

opposite

adjacenttan θ =

Pythagorean Theorem:

a 2 + b

2 = c 2

If a right triangle has legs withmeasures a and b and hypotenusewith measure c, then...

Coordinate Geometry Properties

Distance Formula: d = (x2 – x 1)2 + (y2 – y 1)2

Midpoint: ,

Slope: m =

Point-Slope Formula: (y − y 1) = m (x − x 1)

Slope Intercept Formula: y = mx + b

Standard Equation of a Line: Ax + By = C

y 1 + y 2

2

x 1 + x 2

2

y 2 − y 1x 2 − x 1

FORMULA SHEET

Formulas that you may need to work questions in this document are found below.You may use calculator π or the number 3.14.



Plane Figure Formulas

c

b

h dP = b + c + d

A = bh12

Sum of angle measures = 180(n – 2), where n = number of sides

rC = 2�r A = �r 2

wP = 2l + 2wA = lw

l

s

s P = 4sA = s · s

b

ha P = 2a + 2bA = bh

a

h

b

A = h (a + b)12

P = a + b + c + dc d

Solid Figure Formulas

Euler’s Formula for Polyhedra:

V − E + F = 2

vertices minus edges plus faces = 2

r

h SA = 2�r 2 + 2�rh

V = �r 2h

SA = 2lw + 2lh + 2whV = lwh

w

h

l

r

SA = 4�r 2

V = �r 343

h

r

SA = �r 2 + �r r 2 + h 2

V = �r 2h13

SA = (Area of the base) + (number of sides)(b)( )

V = (Area of the base)(h)13

12h

base

b

b

FORMULA SHEET

Formulas that you may need to work questions in this document are found below.You may use calculator π or the number 3.14.


MODULE 1—Geometric Properties and Reasoning


ASSESSMENT ANCHOR

G.1.1 Properties of Circles, Spheres, and Cylinders

Anchor Descriptor Eligible ContentPA Core

Standards

G.1.1.1 Identify and/or use parts of circles and segments associated with circles, spheres, and cylinders.

G.1.1.1.1 Identify, determine, and/or use the radius, diameter, segment, and/or tangent of a circle.

CC.2.3.HS.A.8

CC.2.3.HS.A.9

CC.2.3.HS.A.13

G.1.1.1.2 Identify, determine, and/or use the arcs, semicircles, sectors, and/or angles of a circle.

G.1.1.1.3 Use chords, tangents, and secants to fi nd missing arc measures or missing segment measures.

G.1.1.1.4 Identify and/or use the properties of a sphere or cylinder.

Sample Exam Questions

Standard G.1.1.1.1

Circle J is inscribed in isosceles trapezoid ABCD, as shown below.

10 cm

20 cm

A B

J

H F

G

E

D C

Points E, F, G, and H are points of tangency. The length of } AB is 10 cm. The length of } DC is 20 cm. What is the length, in cm, of } BC ?

A. 5

B. 10

C. 15

D. 30

Standard G.1.1.1.2

Circle E is shown in the diagram below.

C

B

E

A

D

Line AD is tangent to circle E. The measure of angle DAB is 110°. The measure of minor arc CB is 120°. What is the measure of arc CBA?

A. 220°

B. 240°

C. 250°

D. 260°




Standard G.1.1.1.3

Circle M is shown below.

HL

I

JK

M

36°

Chords } KH , } HI , and } IJ are congruent. What is the measure of C KH ?

A. 72°

B. 90°

C. 96°

D. 108°




Standard G.1.1.1.4

Which of the nets shown below represents a cylinder for all positive values of r ?

A. B.

C. D.

r

r

2πr

r

r

r

2r

πr

r

r

r 2

2π

r

r

r 2

2r




ASSESSMENT ANCHOR

G.1.1 Properties of Circles, Spheres, and Cylinders


Standard G.1.1

A diagram is shown below.

M

K8

N

X

Q

RY

L

7

P3

J

In the diagram, } JM and } JN are tangent to circle X and circle Y.

A. What is the length of } JM ?

length of } JM :

Continued on next page.




Continued. Please refer to the previous page for task explanation.

B. Identify the chord in the diagram.

chord:

C. What is the length of } JP ? Show your work. Explain your reasoning.




Standard G.1.1

A circle is shown below.

S

W

YX

R

T

V

Some information about the circle is listed below.

• @##$ VR is tangent to the circle

• m /VRS = 77°

• m /RST = 27°

• m C SW = 78°

• m C XY = 22°

• m C SY = 78°

A. What is the measure of /XVY?

m /XVY =






B. What is the measure of C RT ?

m C RT =

C. What is the measure of C WT ?

m C WT =

D. What is the measure of /SVR?

m /SVR =




ASSESSMENT ANCHOR

G.1.2 Properties of Polygons and Polyhedra


Standards

G.1.2.1 Recognize and/or apply properties of angles, polygons, and polyhedra.

G.1.2.1.1 Identify and/or use properties of triangles. CC.2.3.8.A.2

CC.2.3.HS.A.3

CC.2.3.HS.A.13G.1.2.1.2 Identify and/or use properties of

quadrilaterals.G.1.2.1.3 Identify and/or use properties of isosceles

and equilateral triangles.G.1.2.1.4 Identify and/or use properties of regular

polygons.G.1.2.1.5 Identify and/or use properties of pyramids

and prisms.


Standard G.1.2.1.1

Acute triangle KLM is shown below.

K

L

25 ft 23 ft

48°M

Which could be the measure of /M?

A. 38°

B. 42°

C. 44°

D. 52°

Standard G.1.2.1.2

Quadrilateral WXYZ is a kite. Which of the following must be true?

A. } WX and } YZ are congruent

B. } WY and } XZ bisect each other

C. } WY and } XZ are perpendicular

D. /WXY and /XYZ are congruent




Standard G.1.2.1.3

A diagram is shown below.

Q

P

VS

R

T U

46°

104°

68°

Which of the triangles must be isosceles?

A. SPR

B. SPQ

C. QTU

D. SQV




Standard G.1.2.1.4

Regular pentagon HIJKL is shown below.

H

I

L

J

K

What is the measure of /HIK?

A. 36º

B. 54º

C. 72º

D. 108º




Standard G.1.2.1.5

In the right rectangular pyramid shown below, x and y are slant heights.

6 in.

y x

3 in.

Which of the following must be true about the relationship between the values of x and y?

A. x = y

B. x > y

C. x < y

D. x2 + y2 = 92




ASSESSMENT ANCHOR

G.1.2 Properties of Polygons and Polyhedra


Standard G.1.2

A craftsman makes a cabinet in the shape of a triangular prism. The top and bottom of the cabinet are congruent isosceles right triangles.

A. Describe the shape needed to build each of the faces of the cabinet.

In order to make production of the cabinets easier, the craftsman wants to design the cabinet so the lateral faces are all congruent figures.

B. Explain why this is not possible.






C. What must be true about the base of a triangular prism in order for the lateral faces to all be congruent figures?

The craftsman is designing the cabinet to fit perfectly against two perpendicular walls.

D. Explain why a cabinet whose lateral faces are all squares cannot fit perfectly against two perpendicular walls.




Standard G.1.2

Kelly notices that in an equilateral triangle, each interior angle measures 60° and in a square, each interior angle measures 90°. She wonders if, each time a side is added to the number of sides in a regular polygon, the measure of each interior angle increases by 30°.

Kelly examines a regular pentagon to see if her hypothesis is correct.

A. What is the measure of each interior angle of a regular pentagon?

interior angle of a regular pentagon:

Kelly decides to examine the ratios of the measures of the interior angles to look for a pattern. She

notices that the ratio of the measure of each interior angle of an equilateral triangle to a square is 2 __ 3

.

B. What is the ratio of the measure of each interior angle of a square to a regular pentagon?

ratio:






Kelly makes a ratio comparing the measure of each interior angle of a regular polygon with n sides to the measure of each interior angle of a regular polygon with n + 1 sides.

C. What is the ratio?

ratio:

D. What is the ratio of the measure of each interior angle of a regular 9-sided polygon to the measure of each interior angle of a regular 10-sided polygon?

ratio:




ASSESSMENT ANCHOR

G.1.3 Congruence, Similarity, and Proofs


Standards

G.1.3.1 Use properties of congruence, correspondence, and similarity in problem-solving settings involving two- and three-dimensional fi gures.

G.1.3.1.1 Identify and/or use properties of congruent and similar polygons or solids.

CC.2.3.HS.A.1

CC.2.3.HS.A.2

CC.2.3.HS.A.5

CC.2.3.HS.A.6G.1.3.1.2 Identify and/or use proportional

relationships in similar fi gures.


Standard G.1.3.1.1

Triangle CDE is similar to triangle FGH. Which relationship must be true?

A. CD + CE } DE

= GH + FH } GH

B. CD } CE

= FH } FG

C. m/ECD + m/CDE + m/GHF = 180°

D. m/ECD + m/CDE = m/HFG + m/GHF

Standard G.1.3.1.2

Trapezoid KLMN is similar to trapezoid ONMP as shown below.

L

K

M

N

P

O

Which relationship must be true?

A. MP } KL

= ON ____ MN

B. LM ___ PO

= NK } NM

C. NK ___ PO

= NM } LM

D. ON ____ KL

= MP ____ MN




ASSESSMENT ANCHOR



Standards

G.1.3.2 Write formal proofs and/or use logic statements to construct or validate arguments.

G.1.3.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction).

CC.2.2.HS.C.9

CC.2.3.HS.A.3

CC.2.3.HS.A.6

CC.2.3.HS.A.8

Sample Exam Question

Standard G.1.3.2.1

The diagram shown below is used in a proof.

F

E D

CG

A B

Given: ABCDEF is a regular hexagon

Prove: AED DBA

A proof is shown below, but statement 7 and reason 7 are missing.

snosaeRstnemetatS

1. ABCDEF is a regular hexagon 1. Given

2. F

AFFE

BCCD

AE BD

ED AB

tneurgnoceranogylopralugerafoselgnA.2C

3. 3. Sides of a regular polygon are congruent

4. AFE BCD 4. Side-angle-side congruence

5. 5. Corresponding parts of congruent triangles arecongruent

6. 6. Sides of a regular polygon are congruent

?.7?.7

8. AED DBA 8. Side-side-side congruence





Standard G.1.3.2.1 (continued)

Which are most likely statement 7 and reason 7 that would complete the proof?

A. statement 7: /AED /ABDreason 7: Angles of a rectangle are congruent

B. statement 7: } AD } BE reason 7: Diagonals of a rectangle are congruent

C. statement 7: } DA } AD reason 7: Refl exive property of congruence

D. statement 7: } EG } GB reason 7: Diagonals of a rectangle bisect each other




ASSESSMENT ANCHOR



Standard G.1.3

In the diagram shown below, JKN | NKM | MKL.

3

J K

N

M

L

4

8

60°

A. What is the length, in units, of } NK ?

length of } NK : units






B. What is the length, in units, of } NM ? Show your work. Explain your reasoning.

C. Prove that the measure of /JKL is 90°.




Standard G.1.3


C

B

A EF

DGiven: ACE is isosceles

Prove: ABE EDA

A proof is shown below, but statement 4 and reason 4 are missing.

ABE EDA

Statements Reasons

1. ACE is isosceles

2. EAC AEC

3.

4.

5.

∠ ∠

1. Given

2. Base angles of isosceles triangles are congruent

AB ED 3. Given

? 4. ?

5. Side-angle-side triangle congruence

A. What could be statement 4 and reason 4 to complete the proof?

statement 4:

reason for statement 4:







Given: ?

Prove: QUR TRU

Q

PU

T

R S

A proof is shown below, but statements 1 and 3 are missing.

QUR TRU

Statements Reasons

1.

∠ ∠

1. Given

2. Corresponding parts of congruent triangles are congruent3. Congruence of segments is reflexive?

4. Side-angle-side triangle congruence

?

2. QR TUQRP TUS

3.

4.

B. What could be statement 1 and statement 3 to complete the proof?

statement 1:

statement 3:



MODULE 2—Coordinate Geometry and Measurement

ASSESSMENT ANCHOR

G.2.1 Coordinate Geometry and Right Triangles


Standards

G.2.1.1 Solve problems involving right triangles.

G.2.1.1.1 Use the Pythagorean theorem to write and/or solve problems involving right triangles.

CC.2.2.HS.C.9

CC.2.3.HS.A.7

G.2.1.1.2 Use trigonometric ratios to write and/or solve problems involving right triangles.


Standard G.2.1.1.1

A kite is shown below.

3

B

E

D

A C

5

3 5√

What is the length of } AC ?

A. 4

B. 6

C. 9

D. 10




Standard G.2.1.1.2

The hypotenuse of each right triangle shown below represents a ladder leaning against a building.

26°

44°

h

15 ft

Which equation can be used to find h, the distance between the base of the building and the point where the shorter ladder touches the building?

A. h = (sin 44°) (15 sin 26°)

B. h = (sin 44°) (15 cos 26°)

C. h = (tan 26°) (15 sin 44°)

D. h = (tan 44°) (15 sin 26°)




ASSESSMENT ANCHOR



Standards

G.2.1.2 Solve problems using analytic geometry.

G.2.1.2.1 Calculate the distance and/or midpoint between two points on a number line or on a coordinate plane.

CC.2.3.8.A.3

CC.2.3.HS.A.11

G.2.1.2.2 Relate slope to perpendicularity and/orparallelism (limit to linear algebraic equations).

G.2.1.2.3 Use slope, distance, and/or midpoint between two points on a coordinate plane to establish properties of a two-dimensional shape.





Standard G.2.1.2.1

The segments on the coordinate plane below represent two parallel roads in a neighborhood. Each unit on

the plane represents 1 } 2 mile.

4 8–8 –4

y

x

8

4

–4

Scale

= mile 12

–8

A new road will be built connecting the midpoints of the two parallel roads. Which is the closest

approximation of the length of the new road connecting the two midpoints?

A. 1.8 miles

B. 3.2 miles

C. 6.4 miles

D. 12.8 miles




Standard G.2.1.2.2

Line p contains the points (9, 7) and (13, 5). Which equation represents a line perpendicular to line p?

A. –2x + y = –11

B. –x – 2y = –2

C. –x + 2y = 5

D. 2x + y = 31

Standard G.2.1.2.3

A map shows flagpoles on a coordinate grid. The flagpoles are at ( –5, 2), ( –5, 6), ( –1, 6), and (2, –1). What type of quadrilateral is formed by the flagpoles on the coordinate grid?

A. kite

B. rectangle

C. rhombus

D. square




Standard G.2.1

A scientist is plotting the circular path of a particle on a coordinate plane for a lab experiment. The scientist knows the path is a perfect circle and that the particle starts at ordered pair (–5, –11). When the particle is halfway around the circle, the particle is at the ordered pair (11, 19).

The segment formed by connecting these two points has the center of the circle as its midpoint.

A. What is the ordered pair that represents the center of the circle?

center of the circle: ( , )

B. What is the length of the radius, in units, of the circle?

radius = units

ASSESSMENT ANCHOR








C. Explain why the particle can never pass through a point with an x-coordinate of 24 as long as it stays on the circular path.

The scientist knows the particle will intersect the line y = 6 twice. The intersection of the particle and the line can be expressed as the ordered pair (x, 6). The value for the x-coordinate of one of the ordered pairs is x ø 19.88.

D. State an approximate value for the other x-coordinate.

x ø




Standard G.2.1

For a social studies project, Darius has to make a map of a neighborhood that could exist in his hometown. He wants to make a park in the shape of a right triangle. He has already planned 2 of the streets that make up 2 sides of his park. The hypotenuse of the park is 3rd Avenue, which goes through the points (–3, 2) and (9, 7) on his map.

One of the legs is Elm Street, which goes through (12, 5) and has a slope of – 2 } 3 . The other leg of the park

will be Spring Parkway and will go through (–3, 2) and intersect Elm Street.

A. What is the slope of Spring Parkway?

slope =

B. What is the length, in units, of 3rd Avenue?

length = units






The variable x represents the length of Spring Parkway in units. The measure of the angle formed by Spring Parkway and 3rd Avenue is approximately 33.69°.

C. Write a trigonometric equation relating the measure of the angle formed by Spring Parkway and 3rd Avenue, the length of Spring Parkway (x), and the length of 3rd Avenue from part B.

equation:

D. Solve for x, the length of Spring Parkway in units.

x = units




ASSESSMENT ANCHOR

G.2.2 Measurements of Two-Dimensional Shapes and Figures


Standards

G.2.2.1 Use and/or compare measurements of angles.

G.2.2.1.1 Use properties of angles formed by intersecting lines to fi nd the measures of missing angles.

CC.2.3.8.A.2

CC.2.3.HS.A.3

G.2.2.1.2 Use properties of angles formed when two parallel lines are cut by a transversal to fi nd the measures of missing angles.


Standard G.2.2.1.1

A figure is shown below.

52°38°

59° x°

What is the value of x?

A. 14

B. 21

C. 31

D. 45




Standard G.2.2.1.2

Parallelogram ABCD is shown below.

A B

CD

130°70°

E

Ray DE passes through the vertex of /ADC. What is the measure of /ADE?

A. 20°

B. 40°

C. 50°

D. 70°




ASSESSMENT ANCHOR



Standards

G.2.2.2 Use and/or develop procedures to determine or describe measures of perimeter, circumference, and/orarea. (May require conversions within the same system.)

G.2.2.2.1 Estimate area, perimeter, or circumference of an irregular fi gure.

CC.2.2.HS.C.1

CC.2.3.HS.A.3

CC.2.3.HS.A.9G.2.2.2.2 Find the measurement of a missing length,

given the perimeter, circumference, or area.

G.2.2.2.3 Find the side lengths of a polygon with a given perimeter to maximize the area of the polygon.

G.2.2.2.4 Develop and/or use strategies to estimate the area of a compound/composite fi gure.

G.2.2.2.5 Find the area of a sector of a circle.


Standard G.2.2.2.1

A diagram of Jacob’s deck is shown below.

5 ft

4 ft

3 ft

7 ft

Jacob’s Deck

Which is the closest approximation of the perimeter of the deck?

A. 29 feet

B. 30 feet

C. 33 feet

D. 34 feet

Standard G.2.2.2.2

A rectangular basketball court has a perimeter of 232 feet. The length of the court is 32 feet greater than the width. What is the width of the basketball court?

A. 42 feet

B. 74 feet

C. 84 feet

D. 100 feet




Standard G.2.2.2.3

Stephen is making a rectangular garden. He has purchased 84 feet of fencing. What length (l ) and width (w) will maximize the area of a garden with a perimeter of 84 feet?

A. l = 21 feetw = 21 feet

B. l = 32 feetw = 10 feet

C. l = 42 feetw = 42 feet

D. l = 64 feetw = 20 feet

Standard G.2.2.2.4

A figure is shown on the grid below.

4 ft

Which expression represents the area of the fi gure?

A. (π(2.5)2 + (3 + 9)5) ft2

B. 1 1 __ 2

π(2.5)2 + 1 } 2 (3 + 9)5 2 ft2

C. 1 1 __ 2

π(2.5)2 + 1 } 2 (3 + 9)5 2 4 ft2

D. 1 1 __ 2

π(2.5)2 + 1 } 2 (3 + 9)5 2 (4•4) ft2

Standard G.2.2.2.5

A circular pizza with a diameter of 18 inches is cut into 8 equal-sized slices. Which is the closest

approximation to the area of 1 slice of pizza?

A. 15.9 square inches

B. 31.8 square inches

C. 56.5 square inches

D. 127.2 square inches




ASSESSMENT ANCHOR



Standards

G.2.2.3 Describe how a change in one dimension of a two-dimensional fi gure affects other measurements of that fi gure.

G.2.2.3.1 Describe how a change in the linear dimension of a fi gure affects its perimeter, circumference, and area (e.g., How does changing the length of the radius of a circle affect the circumference of the circle?).

CC.2.3.HS.A.8

CC.2.3.HS.A.9


Standard G.2.2.3.1

Jack drew a rectangle labeled X with a length of 10 centimeters (cm) and a width of 25 cm. He drew another rectangle labeled Y with a length of 15 cm and the same width as rectangle X. Which is a true statement about the rectangles?

A. The area of rectangle Y will be 3 } 5 times greater

than the area of rectangle X.

B. The area of rectangle Y will be 3 } 2 times greater

than the area of rectangle X.

C. The perimeter of rectangle Y will be 3 } 5 times

greater than the perimeter of rectangle X.

D. The perimeter of rectangle Y will be 3 } 2 times

greater than the perimeter of rectangle X.




ASSESSMENT ANCHOR



Standards

G.2.2.4 Apply probability to practical situations.

G.2.2.4.1 Use area models to fi nd probabilities. CC.2.3.HS.A.14


Standard G.2.2.4.1

Jamal rolls two 6-sided number cubes labeled 1 through 6. Which area model can Jamal use to correctly determine the probability that one of the number cubes will have a 3 facing up and the other will have an even number facing up?

A. B.

C. D.

123456

1 2 3 4 65123456

1 2 3 4 65

123456

1 2 3 4 65123456

1 2 3 4 65




Standard G.2.2.4.1

Michael’s backyard is in the shape of an isosceles trapezoid and has a semicircular patio, as shown in the diagram below.

patio

50 ft

30 ft

65 ft

On a windy fall day, a leaf lands randomly in Michael’s backyard. Which is the closest approximation of the probability that the leaf lands somewhere in the section of the backyard represented by the shaded region in the diagram?

A. 15%

B. 30%

C. 70%

D. 85%




Standard G.2.2

The diagram below shows the dimensions of Tessa’s garden.

210 ft

100 ft 80 ft

40 ft

Tessa’s Garden

A. What is the perimeter, in feet, of Tessa’s garden? Show or explain all your work.

ASSESSMENT ANCHOR








B. What is the area, in square feet, of Tessa’s garden?

area of Tessa’s garden: square feet

Tessa decided that she liked the shape of her garden but wanted to have 2 times the area. She drew a design for a garden with every dimension multiplied by 2.

C. Explain the error in Tessa’s design.




Standard G.2.2

Donovan and Eric are playing in a checkers tournament. Donovan has a 70% chance of winning his game. Eric has a 40% chance of winning his game.

A. What is the difference between the probability that Donovan and Eric both win and the probability that they both lose?

difference:






Donovan and Eric created the probability model shown below to represent all possible outcomes for each of them playing 1 game.

B. Describe the compound event that is represented by the shaded region.

Sari and Keiko are also playing in a checkers tournament. The probability that Sari wins her game is double the probability that Keiko wins her game. The probability Sari wins her game and Keiko loses her game is 0.48.

C. What is the probability that Sari wins her game?

probability Sari wins:




ASSESSMENT ANCHOR

G.2.3 Measurements of Three-Dimensional Shapes and Figures


Standards

G.2.3.1 Use and/or develop procedures to determine or describe measures of surface area and/or volume. (May require conversions within the same system.)

G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

CC.2.3.8.A.1

CC.2.3.HS.A.12

CC.2.3.HS.A.14

G.2.3.1.2 Calculate the volume of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.3 Find the measurement of a missing length, given the surface area or volume.


Standard G.2.3.1.1

A sealed container is shaped like a right cylinder. The exterior height is 80 cm. The exterior diameter of each base is 28 cm. The circumference of each base is approximately 87.92 cm. The longest diagonal is approximately 84.76 cm. The measure of the total exterior surface area of the container can be used to determine the amount of paint needed to cover the container. Which is the closest approximation of the total exterior surface area, including the bases, of the container?

A. 7,209.44 square cm

B. 7,649.04 square cm

C. 8,011.68 square cm

D. 8,264.48 square cm

Standard G.2.3.1.2

A freshwater tank shaped like a rectangular prism has a length of 72 inches, a width of 24 inches, and a height of 25 inches. The tank is filled with water at a constant rate of 5 cubic feet per hour. How long will it take to fill the tank halfway?

A. 2.5 hours

B. 5 hours

C. 12.5 hours

D. 25 hours




Standard G.2.3.1.3

Every locker at a school has a volume of 10.125 cubic feet. The length and width are both 1.5 feet as shown below.

1.5 feet1.5 feet

x feet

Locker

What is the value of x?

A. 3.375

B. 4.5

C. 6.75

D. 7.125




ASSESSMENT ANCHOR



Standards

G.2.3.2 Describe how a change in one dimension of a three-dimensional fi gure affects other measurements of that fi gure.

G.2.3.2.1 Describe how a change in the linear dimension of a fi gure affects its surface area or volume (e.g., How does changing the length of the edge of a cube affect the volume of the cube?).

CC.2.3.HS.A.13


Standard G.2.3.2.1

Anya is wrapping gift boxes in paper. Each gift box is a rectangular prism. The larger box has a length, width, and height twice as large as the smaller box. Which statement shows the relationship between the surface area of the gift boxes?

A. The larger gift box has a surface area 2 times as large as the smaller gift box.

B. The larger gift box has a surface area 4 times as large as the smaller gift box.

C. The larger gift box has a surface area 6 times as large as the smaller gift box.

D. The larger gift box has a surface area 8 times as large as the smaller gift box.




ASSESSMENT ANCHOR



Standard G.2.3

Max is building a spherical model of Earth. He is building his model using 2-inch-long pieces of wood to construct the radius.

The first time he tries to build the model, he uses 3 of the 2-inch pieces of wood end-to-end to make the radius of the model.

A. What is the volume of the model in cubic inches?

volume: cubic inches






In order to purchase the right amount of paint for the outside of the model, Max needs to know the surface area.

B. What is the surface area of the model in square inches?

surface area: square inches

Max decides he wants the model to be larger. He wants the new model to have exactly twice the volume of the original model.

C. Explain why Max cannot make a model that has exactly twice the volume without breaking the 2-inch-long pieces of wood he is using to construct the radius.

Max is going to make a new larger model using n 2-inch-long pieces of wood.

D. How many times greater than the surface area of the original model will the new model be?





Standard G.2.3

An engineer for a storage company is designing cylindrical containers.

A. The first container he designs is a cylinder with a radius of 3 inches and a height of 8 inches. What is the volume of the container in cubic inches?

volume: cubic inches

B. When the surface area is x square inches and the volume is x cubic inches, what is the measure of the height (h) in terms of the radius (r) ?

h =





A cylindrical container has a surface area of x square inches and a volume of x cubic inches. The radius of the cylindrical container is 6 inches.

C. Using your equation from part B, what is the height, in inches, of the cylindrical container?

height: inches

D. What number (x) represents both the surface area, in square inches, and the volume, in cubic inches, of the cylindrical container?

x =


KEYSTONE GEOMETRY ASSESSMENT ANCHORS

KEY TO SAMPLE MULTIPLE-CHOICE ITEMS

Geometry

Eligible Content Key

G.1.1.1.1 C

G.1.1.1.2 D

G.1.1.1.3 C

G.1.1.1.4 A


G.1.2.1.1 C

G.1.2.1.2 C

G.1.2.1.3 D

G.1.2.1.4 C

G.1.2.1.5 B


G.1.3.1.1 C

G.1.3.1.2 D

G.1.3.2.1 C


G.2.1.1.1 D

G.2.1.1.2 D


G.2.1.2.1 B

G.2.1.2.2 A

G.2.1.2.3 A


G.2.2.1.1 C

G.2.2.1.2 A


G.2.2.2.1 C

G.2.2.2.2 A

G.2.2.2.3 A

G.2.2.2.4 D

G.2.2.2.5 B


G.2.2.3.1 B


G.2.2.4.1 (page 42) D

G.2.2.4.1 (page 43) D


G.2.3.1.1 D

G.2.3.1.2 A

G.2.3.1.3 B


G.2.3.2.1 B

Key

ston

e Ex

ams:

Geo

met

ry

Glos

sary

to th

e

Asse

ssm

ent A

ncho

r & E

ligib

le C

onte

nt

The

Keys

tone

Glo

ssar

y in

clud

es t

erm

s an

d de

finiti

ons

asso

ciat

ed w

ith t

he K

eyst

one

Asse

ssm

ent

Anch

ors

and

Elig

ible

Con

tent

. Th

e te

rms

and

defin

ition

s in

clud

ed i

n th

e gl

ossa

ry a

re i

nten

ded

to a

ssis

t Pe

nnsy

lvan

ia

educ

ator

s in

bet

ter

unde

rsta

ndin

g th

e Ke

ysto

ne A

sses

smen

t An

chor

s an

d El

igib

le C

onte

nt. T

he g

loss

ary

does

no

t de

fine

all p

ossi

ble

term

s in

clud

ed o

n an

act

ual K

eyst

one

Exam

, and

it is

not

inte

nded

to

defin

e te

rms

for

use

in c

lass

room

inst

ruct

ion

for a

par

ticul

ar g

rade

leve

l or c

ours

e.

Penn

sylv

ania

Dep

artm

ent o

f Edu

catio

n w

ww

.edu

cati

on.s

tate

.pa.

us

Apri

l 201

4

http://www.education.state.pa.us/

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

2

Apri

l 201

4

Acu

te A

ngle

A

n an

gle

that

mea

sure

s gr

eate

r tha

n 0°

but

less

than

90°

. An

angl

e la

rger

than

a z

ero

angl

e bu

t sm

alle

r th

an a

righ

t ang

le.

Acu

te T

riang

le

A tr

iang

le in

whi

ch e

ach

angl

e m

easu

res

less

than

90°

(i.e

., th

ere

are

thre

e ac

ute

angl

es).

Alti

tude

(of a

Sol

id)

The

shor

test

line

seg

men

t bet

wee

n th

e ba

se a

nd th

e op

posi

te v

erte

x of

a p

yram

id o

r con

e, w

ith o

ne

endp

oint

at t

he v

erte

x. T

he s

horte

st li

ne s

egm

ent b

etw

een

two

base

s of

a p

rism

or c

ylin

der.

The

line

segm

ent i

s pe

rpen

dicu

lar t

o th

e ba

se(s

) of t

he s

olid

. The

alti

tude

may

ext

end

from

eith

er th

e ba

se o

f the

so

lid o

r fro

m th

e pl

ane

exte

ndin

g th

roug

h th

e ba

se. I

n a

right

sol

id, t

he a

ltitu

de c

an b

e fo

rmed

at t

he

cent

er o

f the

bas

e(s)

.

Alti

tude

(of a

Tria

ngle

) A

line

seg

men

t with

one

end

poin

t at a

ver

tex

of th

e tri

angl

e th

at is

per

pend

icul

ar to

the

side

opp

osite

the

verte

x. T

he o

ther

end

poin

t of t

he a

ltitu

de m

ay e

ither

be

on th

e si

de o

f the

tria

ngle

or o

n th

e lin

e ex

tend

ing

thro

ugh

the

side

.

Ana

lytic

Geo

met

ry

The

stud

y of

geo

met

ry u

sing

alg

ebra

(i.e

., po

ints

, lin

es, a

nd s

hape

s ar

e de

scrib

ed in

term

s of

thei

r co

ordi

nate

s, th

en a

lgeb

ra is

use

d to

pro

ve th

ings

abo

ut th

ese

poin

ts, l

ines

, and

sha

pes)

. The

des

crip

tion

of g

eom

etric

figu

res

and

thei

r rel

atio

nshi

ps w

ith a

lgeb

raic

equ

atio

ns o

r vic

e-ve

rsa.

Ang

le

The

incl

inat

ion

betw

een

inte

rsec

ting

lines

, lin

e se

gmen

ts, a

nd/o

r ray

s m

easu

red

in d

egre

es (e

.g.,

a 90

° in

clin

atio

n is

a ri

ght a

ngle

). Th

e fig

ure

is o

ften

repr

esen

ted

by tw

o ra

ys th

at h

ave

a co

mm

on e

ndpo

int.

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

3

Apri

l 201

4

Ang

le B

isec

tor

A li

ne, l

ine

segm

ent,

or ra

y w

hich

cut

s a

give

n an

gle

in h

alf c

reat

ing

two

cong

ruen

t ang

les.

Exa

mpl

e:

Arc

(of a

Circ

le)

Any

con

tinuo

us p

art o

f a c

ircle

bet

wee

n tw

o po

ints

on

the

circ

le.

Are

a Th

e m

easu

re, i

n sq

uare

uni

ts o

r uni

ts2 , o

f the

sur

face

of a

pla

ne fi

gure

(i.e

., th

e nu

mbe

r of s

quar

e un

its it

ta

kes

to c

over

the

figur

e).

Bas

e (T

hree

Dim

ensi

ons)

In

a c

one

or p

yram

id, t

he fa

ce o

f the

figu

re w

hich

is o

ppos

ite th

e ve

rtex.

In a

cyl

inde

r or p

rism

, eith

er o

f th

e tw

o fa

ces

of th

e fig

ure

whi

ch a

re p

aral

lel a

nd c

ongr

uent

.

Bas

e (T

wo

Dim

ensi

ons)

In

an

isos

cele

s tri

angl

e, th

e si

de o

f the

figu

re w

hich

is a

djac

ent t

o th

e co

ngru

ent a

ngle

s. In

a tr

apez

oid,

ei

ther

of t

he p

aral

lel s

ides

of t

he fi

gure

.

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

4

Apri

l 201

4

Cen

tral

Ang

le (o

f a C

ircle

) A

n an

gle

who

se v

erte

x is

at t

he c

ente

r of a

circ

le a

nd w

hose

sid

es a

re ra

dii o

f tha

t circ

le. E

xam

ple:

∠

PO

Q w

ith v

erte

x O

Cen

tral

Ang

le (o

f a

Reg

ular

Pol

ygon

) A

n an

gle

who

se v

erte

x is

at t

he c

ente

r of t

he p

olyg

on a

nd w

hose

sid

es in

ters

ect t

he re

gula

r pol

ygon

at

adja

cent

ver

tices

.

Cen

troi

d A

poi

nt o

f con

curre

ncy

for a

tria

ngle

that

can

be

foun

d at

the

inte

rsec

tion

of th

e th

ree

med

ians

of t

he

trian

gle.

Thi

s po

int i

s al

so th

e ce

nter

of b

alan

ce o

f a tr

iang

le w

ith u

nifo

rm m

ass.

It is

som

etim

es re

ferre

d to

as

the

“cen

ter o

f gra

vity

.” E

xam

ple:

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

5

Apri

l 201

4

Cho

rd

A li

ne s

egm

ent w

hose

two

endp

oint

s ar

e on

the

perim

eter

of a

circ

le. A

par

ticul

ar ty

pe o

f cho

rd th

at

pass

es th

roug

h th

e ce

nter

of t

he c

ircle

is c

alle

d a

diam

eter

. A c

hord

is p

art o

f a s

ecan

t of t

he c

ircle

. E

xam

ple:

Circ

le

A tw

o-di

men

sion

al fi

gure

for w

hich

all

poin

ts a

re th

e sa

me

dist

ance

from

its

cent

er. I

nfor

mal

ly, a

per

fect

ly

roun

d sh

ape.

The

circ

le is

nam

ed fo

r its

cen

ter p

oint

. Exa

mpl

e:

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

6

Apri

l 201

4

Circ

umce

nter

A

poi

nt o

f con

curre

ncy

for a

tria

ngle

that

can

be

foun

d at

the

inte

rsec

tion

of th

e th

ree

perp

endi

cula

r bi

sect

ors

of th

e tri

angl

e. T

his

poin

t is

also

the

cent

er o

f a c

ircle

that

can

be

circ

umsc

ribed

abo

ut th

e tri

angl

e. E

xam

ple:

Circ

umfe

renc

e (o

f a

Circ

le)

The

tota

l mea

sure

d di

stan

ce a

roun

d th

e ou

tsid

e of

a c

ircle

. The

circ

le’s

per

imet

er. M

ore

form

ally

, a

com

plet

e ci

rcul

ar a

rc.

Circ

umsc

ribed

Circ

le

A c

ircle

aro

und

a po

lygo

n su

ch th

at e

ach

verte

x of

the

poly

gon

is a

poi

nt o

n th

e ci

rcle

.

Col

inea

r Tw

o or

mor

e po

ints

that

lie

on th

e sa

me

line.

Com

posi

te (C

ompo

und)

Fi

gure

(Sha

pe)

A fi

gure

mad

e fro

m tw

o or

mor

e ge

omet

ric fi

gure

s (i.

e., f

rom

“sim

pler

” fig

ures

).

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

7

Apri

l 201

4

Con

e A

thre

e-di

men

sion

al fi

gure

with

a s

ingl

e ci

rcul

ar b

ase

and

one

verte

x. A

cur

ved

surfa

ce c

onne

cts

the

base

and

the

verte

x. T

he s

horte

st d

ista

nce

from

the

base

to th

e ve

rtex

is c

alle

d th

e al

titud

e. If

the

altit

ude

goes

thro

ugh

the

cent

er o

f the

bas

e, th

e co

ne is

cal

led

a “ri

ght c

one”

; oth

erw

ise,

it is

cal

led

an

“obl

ique

con

e.” U

nles

s ot

herw

ise

spec

ified

, it m

ay b

e as

sum

ed a

ll co

nes

are

right

con

es. E

xam

ple:

co

ne

Con

grue

nt F

igur

es

Two

or m

ore

figur

es h

avin

g th

e sa

me

shap

e an

d si

ze (i

.e.,

mea

sure

). A

ngle

s ar

e co

ngru

ent i

f the

y ha

ve

the

sam

e m

easu

re. L

ine

segm

ents

are

con

grue

nt if

they

hav

e th

e sa

me

leng

th. T

wo

or m

ore

shap

es o

r so

lids

are

said

to b

e co

ngru

ent i

f the

y ar

e “id

entic

al” i

n ev

ery

way

exc

ept f

or p

ossi

bly

thei

r pos

ition

. W

hen

cong

ruen

t fig

ures

are

nam

ed, t

heir

corre

spon

ding

ver

tices

are

list

ed in

the

sam

e or

der (

e.g.

, if

trian

gle

AB

C is

con

grue

nt to

tria

ngle

XYZ

, the

n ve

rtex

C c

orre

spon

ds to

ver

tex

Z).

Con

vers

ion

The

proc

ess

of c

hang

ing

the

form

of a

mea

sure

men

t, bu

t not

its

valu

e (e

.g.,

4 in

ches

con

verts

to 1 3

foot

;

4 sq

uare

met

ers

conv

erts

to 0

.000

004

squa

re k

ilom

eter

s; 4

cub

ic fe

et c

onve

rts to

6,9

12 c

ubic

inch

es).

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

8

Apri

l 201

4

Coo

rdin

ate

Plan

e A

pla

ne fo

rmed

by

perp

endi

cula

r num

ber l

ines

. The

hor

izon

tal n

umbe

r lin

e is

the

x-ax

is, a

nd th

e ve

rtica

l nu

mbe

r lin

e is

the

y-ax

is. T

he p

oint

whe

re th

e ax

es m

eet i

s ca

lled

the

orig

in. E

xam

ple:

co

ordi

nate

pla

ne

Coo

rdin

ates

Th

e or

dere

d pa

ir of

num

bers

giv

ing

the

loca

tion

or p

ositi

on o

f a p

oint

on

a co

ordi

nate

pla

ne. T

he o

rder

ed

pairs

are

writ

ten

in p

aren

thes

es (e

.g.,

(x, y

) whe

re th

e x-

coor

dina

te is

the

first

num

ber i

n an

ord

ered

pai

r an

d re

pres

ents

the

horiz

onta

l pos

ition

of a

n ob

ject

in a

coo

rdin

ate

plan

e an

d th

e y-

coor

dina

te is

the

seco

nd n

umbe

r in

an o

rder

ed p

air a

nd re

pres

ents

the

verti

cal p

ositi

on o

f an

obje

ct in

a c

oord

inat

e pl

ane)

.

Cop

lana

r Tw

o or

mor

e fig

ures

that

lie

in th

e sa

me

plan

e.

http://www.mathwords.com/p/point.htm

http://www.mathwords.com/c/coordinate_plane.htm

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

9

Apri

l 201

4

Cor

resp

ondi

ng A

ngle

s P

airs

of a

ngle

s ha

ving

the

sam

e re

lativ

e po

sitio

n in

geo

met

ric fi

gure

s (i.

e., a

ngle

s on

the

sam

e si

de o

f a

trans

vers

al fo

rmed

whe

n tw

o pa

ralle

l lin

es a

re in

ters

ecte

d by

the

trans

vers

al; f

our s

uch

pairs

are

form

ed,

and

the

angl

es w

ithin

the

pairs

are

equ

al to

eac

h ot

her).

Cor

resp

ondi

ng a

ngle

s ar

e eq

ual i

n m

easu

re.

Cor

resp

ondi

ng P

arts

Tw

o pa

rts (a

ngle

s, s

ides

, or v

ertic

es) h

avin

g th

e sa

me

rela

tive

posi

tion

in c

ongr

uent

or s

imila

r fig

ures

. W

hen

cong

ruen

t or s

imila

r fig

ures

are

nam

ed, t

heir

corre

spon

ding

ver

tices

are

list

ed in

the

sam

e or

der

(e.g

., if

trian

gle

AB

C is

sim

ilar t

o tri

angl

e XY

Z, th

en v

erte

x C

cor

resp

onds

to v

erte

x Z)

. See

als

o co

rresp

ondi

ng a

ngle

s an

d co

rresp

ondi

ng s

ides

.

Cor

resp

ondi

ng S

ides

Tw

o si

des

havi

ng th

e sa

me

rela

tive

posi

tion

in tw

o di

ffere

nt fi

gure

s. If

the

figur

es a

re c

ongr

uent

or

sim

ilar,

the

side

s m

ay b

e, re

spec

tivel

y, e

qual

in le

ngth

or p

ropo

rtion

al.

Cos

ine

(of a

n An

gle)

A

trig

onom

etric

ratio

with

in a

righ

t tria

ngle

. The

ratio

is th

e le

ngth

of t

he le

g ad

jace

nt to

the

angl

e to

the

leng

th o

f the

hyp

oten

use

of th

e tri

angl

e.

cosi

ne o

f an

angl

e =

leng

th o

f adj

acen

t leg

leng

th o

f hyp

oten

use

Cub

e A

thre

e-di

men

sion

al fi

gure

(e.g

., a

rect

angu

lar s

olid

or p

rism

) hav

ing

six

cong

ruen

t squ

are

face

s.

Exa

mpl

e:

cu

be

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

10

Ap

ril 2

014

Cyl

inde

r A

thre

e-di

men

sion

al fi

gure

with

two

circ

ular

bas

es th

at a

re p

aral

lel a

nd c

ongr

uent

and

join

ed b

y st

raig

ht

lines

cre

atin

g a

late

ral s

urfa

ce th

at is

cur

ved.

The

dis

tanc

e be

twee

n th

e ba

ses

is c

alle

d an

alti

tude

. If t

he

altit

ude

goes

thro

ugh

the

cent

er o

f the

bas

es, t

he c

ylin

der i

s ca

lled

a “ri

ght c

ylin

der”;

oth

erw

ise,

it is

ca

lled

an “o

bliq

ue c

ylin

der.”

Unl

ess

othe

rwis

e sp

ecifi

ed, i

t may

be

assu

med

all

cylin

ders

are

righ

t cy

linde

rs. E

xam

ple:

cy

linde

r

Deg

ree

A u

nit o

f ang

le m

easu

re e

qual

to

1 360

of a

com

plet

e re

volu

tion.

The

re a

re 3

60 d

egre

es in

a c

ircle

. The

sym

bol f

or d

egre

e is

° (e

.g.,

45°

is re

ad “4

5 de

gree

s”).

Dia

gona

l A

ny li

ne s

egm

ent,

othe

r tha

n a

side

or e

dge,

with

in a

pol

ygon

or p

olyh

edro

n th

at c

onne

cts

one

verte

x w

ith a

noth

er v

erte

x.

http://www.mathwords.com/m/measure_of_an_angle.htm

http://www.mathwords.com/c/circle.htm

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

11

Ap

ril 2

014

Dia

met

er (o

f a C

ircle

) A

line

seg

men

t tha

t has

end

poin

ts o

n a

circ

le a

nd p

asse

s th

roug

h th

e ce

nter

of t

he c

ircle

. It i

s th

e lo

nges

t ch

ord

in a

circ

le. I

t div

ides

the

circ

le in

hal

f. Ex

ampl

e:

di

amet

er

Dire

ct P

roof

Th

e tru

th o

r val

idity

of a

giv

en s

tate

men

t sho

wn

by a

stra

ight

forw

ard

com

bina

tion

of e

stab

lishe

d fa

cts

(e.g

., ex

istin

g ax

iom

s, d

efin

ition

s, th

eore

ms)

, with

out m

akin

g an

y fu

rther

ass

umpt

ions

(i.e

., a

sequ

ence

of

sta

tem

ents

sho

win

g th

at if

one

thin

g is

true

, the

n so

met

hing

follo

win

g fro

m it

is a

lso

true)

.

Dis

tanc

e be

twee

n Tw

o Po

ints

Th

e sp

ace

show

ing

how

far a

part

two

poin

ts a

re (i

.e.,

the

shor

test

leng

th b

etw

een

them

).

Edge

Th

e lin

e se

gmen

t whe

re tw

o fa

ces

of a

pol

yhed

ron

mee

t (e.

g., a

rect

angu

lar p

rism

has

12

edge

s). T

he

endp

oint

s of

an

edge

are

ver

tices

of t

he p

olyh

edro

n.

Endp

oint

A

poi

nt th

at m

arks

the

begi

nnin

g or

the

end

of a

line

seg

men

t; a

poin

t tha

t mar

ks th

e be

ginn

ing

of a

ray.

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

12

Ap

ril 2

014

Equi

late

ral T

riang

le

A tr

iang

le w

here

all

side

s ar

e th

e sa

me

leng

th (i

.e.,

the

side

s ar

e co

ngru

ent).

Eac

h of

the

angl

es in

an

equi

late

ral t

riang

le is

60°

. Thu

s, th

e tri

angl

e is

als

o “e

quia

ngul

ar.”

Exam

ple:

eq

uila

tera

l tria

ngle

ABC

Exte

rior A

ngle

A

n an

gle

form

ed b

y a

side

of a

pol

ygon

and

an

exte

nsio

n of

an

adja

cent

sid

e. T

he m

easu

re o

f the

ex

terio

r ang

le is

sup

plem

enta

ry to

the

mea

sure

of t

he in

terio

r ang

le a

t tha

t ver

tex.

Face

A

pla

ne fi

gure

or f

lat s

urfa

ce th

at m

akes

up

one

side

of a

thre

e-di

men

sion

al fi

gure

or s

olid

figu

re. T

wo

face

s m

eet a

t an

edge

, thr

ee o

r mor

e fa

ces

mee

t at a

ver

tex

(e.g

., a

cube

has

6 fa

ces)

. See

als

o la

tera

l fa

ce.

Figu

re

Any

com

bina

tion

of p

oint

s, li

nes,

rays

, lin

e se

gmen

ts, a

ngle

s, p

lane

s, o

r cur

ves

in tw

o or

thre

e di

men

sion

s. F

orm

ally

, it i

s an

y se

t of p

oint

s on

a p

lane

or i

n sp

ace.

http://www.mathwords.com/s/set.htm

http://www.mathwords.com/t/three_dimensions.htm

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

13

Ap

ril 2

014

Hyp

oten

use

The

long

est s

ide

of a

righ

t tria

ngle

(i.e

., th

e si

de a

lway

s op

posi

te th

e rig

ht a

ngle

). E

xam

ple:

rig

ht tr

iang

le A

BC

, with

hyp

oten

use

AC

Ince

nter

A

poi

nt o

f con

curre

ncy

for a

tria

ngle

that

can

be

foun

d at

the

inte

rsec

tion

of th

e th

ree

angl

e bi

sect

ors

of

the

trian

gle.

Thi

s po

int i

s al

so th

e ce

nter

of a

circ

le th

at c

an b

e in

scrib

ed w

ithin

the

trian

gle.

Exa

mpl

e:

Indi

rect

Pro

of

A s

et o

f sta

tem

ents

in w

hich

a fa

lse

assu

mpt

ion

is m

ade.

Usi

ng tr

ue o

r val

id a

rgum

ents

, a s

tate

men

t is

arriv

ed a

t, bu

t it i

s cl

early

wro

ng, s

o th

e or

igin

al a

ssum

ptio

n m

ust h

ave

been

wro

ng. S

ee a

lso

proo

f by

cont

radi

ctio

n.

Insc

ribed

Circ

le

A c

ircle

with

in a

pol

ygon

suc

h th

at e

ach

side

of t

he p

olyg

on is

tang

ent t

o th

e ci

rcle

.

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

14

Ap

ril 2

014

Inte

rior A

ngle

A

n an

gle

form

ed b

y tw

o ad

jace

nt s

ides

of a

pol

ygon

. The

com

mon

end

poin

t of t

he s

ides

form

the

verte

x of

the

angl

e, w

ith th

e in

clin

atio

n of

mea

sure

bei

ng o

n th

e in

side

of t

he p

olyg

on.

Inte

rsec

ting

Line

s Tw

o lin

es th

at c

ross

or m

eet e

ach

othe

r. Th

ey a

re c

opla

nar,

have

onl

y on

e po

int i

n co

mm

on, h

ave

slop

es th

at a

re n

ot e

qual

, are

not

par

alle

l, an

d fo

rm a

ngle

s at

the

poin

t of i

nter

sect

ion.

Irreg

ular

Fig

ure

A fi

gure

that

is n

ot re

gula

r; no

t all

side

s an

d/or

ang

les

are

cong

ruen

t.

Isos

cele

s Tr

iang

le

A tr

iang

le th

at h

as a

t lea

st tw

o co

ngru

ent s

ides

. The

third

sid

e is

cal

led

the

base

. The

ang

les

oppo

site

th

e eq

ual s

ides

are

als

o co

ngru

ent.

Exa

mpl

e:

is

osce

les

trian

gle

ABC

, with

bas

e B

C

Late

ral F

ace

Any

face

or s

urfa

ce o

f a th

ree-

dim

ensi

onal

figu

re o

r sol

id th

at is

not

a b

ase.

http://www.mathwords.com/b/base.htm

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

15

Ap

ril 2

014

Leg

(of a

Rig

ht T

riang

le)

Eith

er o

f the

two

side

s th

at fo

rm a

righ

t ang

le in

a ri

ght t

riang

le. I

t is

one

of th

e tw

o sh

orte

r sid

es o

f the

tri

angl

e an

d al

way

s op

posi

te a

n ac

ute

angl

e. It

is n

ot th

e hy

pote

nuse

. Exa

mpl

e:

rig

ht tr

iang

le A

BC

, with

legs

AB

and

BC

Line

A

figu

re w

ith o

nly

one

dim

ensi

on—

leng

th (n

o w

idth

or h

eigh

t). A

stra

ight

pat

h ex

tend

ing

in b

oth

dire

ctio

ns w

ith n

o en

dpoi

nts.

It is

con

side

red

“nev

er e

ndin

g.” F

orm

ally

, it i

s an

infin

ite s

et o

f con

nect

ed

poin

ts (i

.e.,

a se

t of p

oint

s so

clo

sely

set

dow

n th

ere

are

no g

aps

or s

pace

s be

twee

n th

em).

The

line

AB

is w

ritte

n , w

here

A a

nd B

are

two

poin

ts th

roug

h w

hich

the

line

pass

es. E

xam

ple:

lin

e A

B (

)

Line

Seg

men

t A

par

t or p

iece

of a

line

or r

ay w

ith tw

o fix

ed e

ndpo

ints

. For

mal

ly, i

t is

the

two

endp

oint

s an

d al

l poi

nts

betw

een

them

. The

line

seg

men

t AB

is w

ritte

n A

B, w

here

A a

nd B

are

the

endp

oint

s of

the

line

segm

ent.

Exa

mpl

e:

lin

e se

gmen

t AB

(A

B)

http://www.mathwords.com/a/acute_angle.htm

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

16

Ap

ril 2

014

Line

ar M

easu

rem

ent

A m

easu

rem

ent t

aken

in a

stra

ight

line

.

Logi

c St

atem

ent

(Pro

posi

tion)

A

sta

tem

ent e

xam

ined

for i

ts tr

uthf

ulne

ss (i

.e.,

prov

ed tr

ue o

r fal

se).

Med

ian

(of a

Tria

ngle

) A

line

seg

men

t with

one

end

poin

t at t

he v

erte

x of

a tr

iang

le a

nd th

e ot

her e

ndpo

int a

t the

mid

poin

t of t

he

side

opp

osite

the

verte

x.

Mid

poin

t Th

e po

int h

alf-w

ay b

etw

een

two

give

n po

ints

(i.e

., it

divi

des

or s

plits

a li

ne s

egm

ent i

nto

two

cong

ruen

t lin

e se

gmen

ts).

Obt

use

Ang

le

An

angl

e th

at m

easu

res

mor

e th

an 9

0° b

ut le

ss th

an 1

80°.

An

angl

e la

rger

than

a ri

ght a

ngle

but

sm

alle

r th

an a

stra

ight

ang

le.

Obt

use

Tria

ngle

A

tria

ngle

with

one

ang

le th

at m

easu

res

mor

e th

an 9

0° (i

.e.,

it ha

s on

e ob

tuse

ang

le a

nd tw

o ac

ute

angl

es).

Ord

ered

Pai

r A

pai

r of n

umbe

rs, (

x, y

), w

ritte

n in

a p

artic

ular

ord

er th

at in

dica

tes

the

posi

tion

of a

poi

nt o

n a

coor

dina

te

plan

e. T

he fi

rst n

umbe

r, x,

repr

esen

ts th

e x-

coor

dina

te a

nd is

the

num

ber o

f uni

ts le

ft or

righ

t fro

m th

e or

igin

; the

sec

ond

num

ber,

y, re

pres

ents

the

y-co

ordi

nate

and

is th

e nu

mbe

r of u

nits

up

or d

own

from

the

orig

in.

Orig

in

The

poin

t (0,

0) o

n a

coor

dina

te p

lane

. It i

s th

e po

int o

f int

erse

ctio

n fo

r the

x-a

xis

and

the

y-ax

is.

javascript:%20new_definition(link2)


K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

17

Ap

ril 2

014

Ort

hoce

nter

A

poi

nt o

f con

curre

ncy

for a

tria

ngle

that

can

be

foun

d at

the

inte

rsec

tion

of th

e th

ree

altit

udes

of t

he

trian

gle.

Exa

mpl

e:

Para

llel (

Bas

es)

Two

base

s of

a th

ree-

dim

ensi

onal

figu

re th

at li

e in

par

alle

l pla

nes.

All

altit

udes

bet

wee

n th

e ba

ses

are

cong

ruen

t.

Para

llel (

Line

s or

Lin

e Se

gmen

ts)

Two

dist

inct

line

s th

at a

re in

the

sam

e pl

ane

and

neve

r int

erse

ct. O

n a

coor

dina

te g

rid, t

he li

nes

have

the

sam

e sl

ope

but d

iffer

ent y

-inte

rcep

ts. T

hey

are

alw

ays

the

sam

e di

stan

ce a

part

from

eac

h ot

her.

Par

alle

l lin

e se

gmen

ts a

re s

egm

ents

of p

aral

lel l

ines

. The

sym

bol f

or p

aral

lel i

s ||

(e.g

., A

B|| C

D is

read

“lin

e se

gmen

t AB

is p

aral

lel t

o lin

e se

gmen

t CD

”).

Para

llel (

Plan

es)

Two

dist

inct

pla

nes

that

nev

er in

ters

ect a

nd a

re a

lway

s th

e sa

me

dist

ance

apa

rt.

Para

llel (

Side

s)

Two

side

s of

a tw

o-di

men

sion

al fi

gure

that

lie

on p

aral

lel l

ines

.

http://www.mathwords.com/d/distinct.htm

http://www.mathwords.com/s/slope_of_a_line.htm

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

18

Ap

ril 2

014

Para

llelo

gram

A

qua

drila

tera

l who

se o

ppos

ite s

ides

are

par

alle

l and

con

grue

nt (i

.e.,

ther

e ar

e tw

o pa

irs o

f par

alle

l si

des)

. Ofte

n on

e pa

ir of

thes

e op

posi

te s

ides

is lo

nger

than

the

othe

r pai

r. O

ppos

ite a

ngle

s ar

e al

so

cong

ruen

t, an

d th

e di

agon

als

bise

ct e

ach

othe

r. E

xam

ple:

pa

ralle

logr

am

Perim

eter

Th

e to

tal d

ista

nce

arou

nd a

clo

sed

figur

e. F

or a

pol

ygon

, it i

s th

e su

m o

f the

leng

ths

of it

s si

des.

Perp

endi

cula

r Tw

o lin

es, s

egm

ents

, or r

ays

that

inte

rsec

t, cr

oss,

or m

eet t

o fo

rm a

90°

or r

ight

ang

le. T

he p

rodu

ct o

f th

eir s

lope

s is

– 1 (i.

e., t

heir

slop

es a

re “n

egat

ive

reci

proc

als”

of e

ach

othe

r). T

he s

ymbo

l for

pe

rpen

dicu

lar i

s ⊥

(e.g

., ⊥

AB

CD

is re

ad “l

ine

segm

ent A

B is

per

pend

icul

ar to

line

seg

men

t CD

”). B

y de

finiti

on, t

he tw

o le

gs o

f a ri

ght t

riang

le a

re p

erpe

ndic

ular

to e

ach

othe

r.

Perp

endi

cula

r Bis

ecto

r A

line

that

inte

rsec

ts a

line

seg

men

t at i

ts m

idpo

int a

nd a

t a ri

ght a

ngle

.

π (P

i) Th

e ra

tio o

f the

circ

umfe

renc

e of

a c

ircle

to it

s di

amet

er. I

t is

3.14

1592

65…

to 1

or s

impl

y th

e va

lue

3.14

1592

65…

. It c

an a

lso

be u

sed

to re

late

the

radi

us o

f a c

ircle

to th

e ci

rcle

’s a

rea.

It is

ofte

n

appr

oxim

ated

usi

ng e

ither

3.1

4 or

22 7.

http://www.mathwords.com/s/side_of_a_polygon.htm

http://www.mathwords.com/s/slope_of_a_line.htm

http://www.mathwords.com/n/negative_reciprocal.htm

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

19

Ap

ril 2

014

Plan

e A

set

of p

oint

s th

at fo

rms

a fla

t sur

face

that

ext

ends

infin

itely

in a

ll di

rect

ions

. It h

as n

o he

ight

.

Plot

ting

Poin

ts

To p

lace

poi

nts

on a

coo

rdin

ate

plan

e us

ing

the

x-co

ordi

nate

s an

d y-

coor

dina

tes

of th

e gi

ven

poin

ts.

Poin

t A

figu

re w

ith n

o di

men

sion

s—it

has

no le

ngth

, wid

th, o

r hei

ght.

It is

gen

eral

ly in

dica

ted

with

a s

ingl

e do

t an

d is

labe

led

with

a s

ingl

e le

tter o

r an

orde

red

pair

on a

coo

rdin

ate

plan

e. E

xam

ple:

●P

poin

t P

Poly

gon

A c

lose

d pl

ane

figur

e m

ade

up o

f thr

ee o

r mor

e lin

e se

gmen

ts (i

.e.,

a un

ion

of li

ne s

egm

ents

con

nect

ed

end

to e

nd s

uch

that

eac

h se

gmen

t int

erse

cts

exac

tly tw

o ot

hers

at i

ts e

ndpo

ints

); le

ss fo

rmal

ly, a

flat

sh

ape

with

stra

ight

sid

es. T

he n

ame

of a

pol

ygon

des

crib

es th

e nu

mbe

r of s

ides

/ang

les

(e.g

., tri

angl

e ha

s th

ree

side

s/an

gles

, a q

uadr

ilate

ral h

as fo

ur, a

pen

tago

n ha

s fiv

e, e

tc.).

Exa

mpl

es:

po

lygo

ns

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

20

Ap

ril 2

014

Poly

hedr

on

A th

ree-

dim

ensi

onal

figu

re o

r sol

id w

hose

flat

face

s ar

e al

l pol

ygon

s w

here

all

edge

s ar

e lin

e se

gmen

ts.

It ha

s no

cur

ved

surfa

ces

or e

dges

. The

plu

ral i

s “p

olyh

edra

.” E

xam

ples

:

po

lyhe

dra

Pris

m

A th

ree-

dim

ensi

onal

figu

re o

r pol

yhed

ron

that

has

two

cong

ruen

t and

par

alle

l fac

es th

at a

re p

olyg

ons

calle

d ba

ses.

The

rem

aini

ng fa

ces,

cal

led

late

ral f

aces

, are

par

alle

logr

ams

(ofte

n re

ctan

gles

). If

the

late

ral f

aces

are

rect

angl

es, t

he p

rism

is c

alle

d a

“righ

t pris

m”;

othe

rwis

e, it

is c

alle

d an

“obl

ique

pris

m.”

Unl

ess

othe

rwis

e sp

ecifi

ed, i

t may

be

assu

med

all

pris

ms

are

right

pris

ms.

Pris

ms

are

nam

ed b

y th

e sh

ape

of th

eir b

ases

. Exa

mpl

es:

http://www.mathwords.com/l/line_segment.htm

http://www.mathwords.com/s/surface.htm

http://www.mathwords.com/e/edge_polyhedron.htm

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

21

Ap

ril 2

014

Proo

f by

Con

trad

ictio

n A

set

of s

tate

men

ts u

sed

to d

eter

min

e th

e tru

th o

f a p

ropo

sitio

n by

sho

win

g th

at th

e pr

opos

ition

bei

ng

untru

e w

ould

impl

y a

cont

radi

ctio

n (i.

e., o

ne a

ssum

es th

at w

hat i

s tru

e is

not

true

, the

n, e

vent

ually

one

di

scov

ers

som

ethi

ng th

at is

cle

arly

not

true

; whe

n so

met

hing

is n

ot n

ot-tr

ue, t

hen

it is

true

). It

is

som

etim

es c

alle

d th

e “la

w o

f dou

ble

nega

tion.

”

Prop

ortio

nal R

elat

ions

hip

A re

latio

nshi

p be

twee

n tw

o eq

ual r

atio

s. It

is o

ften

used

in p

robl

em-s

olvi

ng s

ituat

ions

invo

lvin

g si

mila

r fig

ures

.

Pyra

mid

A

thre

e-di

men

sion

al fi

gure

or p

olyh

edro

n w

ith a

sin

gle

poly

gon

base

and

tria

ngul

ar fa

ces

that

mee

t at a

si

ngle

poi

nt o

r ver

tex.

The

face

s th

at m

eet a

t the

ver

tex

are

calle

d la

tera

l fac

es. T

here

is th

e sa

me

num

ber o

f lat

eral

face

s as

ther

e ar

e si

des

of th

e ba

se. T

he s

horte

st d

ista

nce

from

the

base

to th

e ve

rtex

is c

alle

d th

e al

titud

e. If

the

altit

ude

goes

thro

ugh

the

cent

er o

f the

bas

e, th

e py

ram

id is

cal

led

a “ri

ght

pyra

mid

”; ot

herw

ise,

it is

cal

led

an “o

bliq

ue p

yram

id.”

Unl

ess

othe

rwis

e sp

ecifi

ed, i

t may

be

assu

med

all

pyra

mid

s ar

e rig

ht p

yram

ids.

A p

yram

id is

nam

ed fo

r the

sha

pe o

f its

bas

e (e

.g.,

trian

gula

r pyr

amid

or

squa

re p

yram

id).

Exa

mpl

e:

Pyth

agor

ean

Theo

rem

A

form

ula

for f

indi

ng th

e le

ngth

of a

sid

e of

a ri

ght t

riang

le w

hen

the

leng

ths

of tw

o si

des

are

give

n. It

is

a2 + b

2 = c

2 , whe

re a

and

b a

re th

e le

ngth

s of

the

legs

of a

righ

t tria

ngle

and

c is

the

leng

th o

f the

hy

pote

nuse

.

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

22

Ap

ril 2

014

Qua

drila

tera

l A

four

-sid

ed p

olyg

on. I

t can

be

regu

lar o

r irre

gula

r. Th

e m

easu

res

of it

s fo

ur in

terio

r ang

les

alw

ays

add

up to

360

°.

Rad

ius

(of a

Circ

le)

A li

ne s

egm

ent t

hat h

as o

ne e

ndpo

int a

t the

cen

ter o

f the

circ

le a

nd th

e ot

her e

ndpo

int o

n th

e ci

rcle

. It i

s th

e sh

orte

st d

ista

nce

from

the

cent

er o

f a c

ircle

to a

ny p

oint

on

the

circ

le. I

t is

half

the

leng

th o

f the

di

amet

er. T

he p

lura

l is

“radi

i.” E

xam

ple:

Ray

A

par

t or p

iece

of a

line

with

one

fixe

d en

dpoi

nt. F

orm

ally

, it i

s th

e en

dpoi

nt a

nd a

ll po

ints

in o

ne

dire

ctio

n. T

he ra

y A

B is

writ

ten

, whe

re A

is a

n en

dpoi

nt o

f the

ray

that

pas

ses

thro

ugh

poin

t B.

Exa

mpl

e:

ra

y A

B (

)

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

23

Ap

ril 2

014

Rec

tang

ular

Pris

m

A th

ree-

dim

ensi

onal

figu

re o

r pol

yhed

ron

whi

ch h

as tw

o co

ngru

ent a

nd p

aral

lel r

ecta

ngul

ar b

ases

. In

form

ally

, it i

s a

“box

sha

pe” i

n th

ree

dim

ensi

ons.

Exa

mpl

e:

Reg

ular

Pol

ygon

A

pol

ygon

with

sid

es a

ll th

e sa

me

leng

th a

nd a

ngle

s al

l the

sam

e si

ze (i

.e.,

all s

ides

are

con

grue

nt o

r eq

uila

tera

l, an

d al

l ang

les

are

cong

ruen

t or e

quia

ngul

ar).

Exa

mpl

e:

re

gula

r pol

ygon

Rig

ht A

ngle

A

n an

gle

that

mea

sure

s ex

actly

90°

.

http://www.mathwords.com/t/three_dimensions.htm

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

24

Ap

ril 2

014

Rig

ht T

riang

le

A tr

iang

le w

ith o

ne a

ngle

that

mea

sure

s 90

° (i.

e., i

t has

one

righ

t ang

le a

nd tw

o ac

ute

angl

es).

The

side

op

posi

te th

e rig

ht a

ngle

is c

alle

d th

e hy

pote

nuse

and

the

two

othe

r sid

es a

re c

alle

d th

e le

gs.

rig

ht tr

iang

le A

BC

Scal

ene

Tria

ngle

A

tria

ngle

that

has

no

cong

ruen

t sid

es (i

.e.,

the

thre

e si

des

all h

ave

diffe

rent

leng

ths)

. The

tria

ngle

als

o ha

s no

con

grue

nt a

ngle

s (i.

e., t

he th

ree

angl

es a

ll ha

ve d

iffer

ent m

easu

res)

.

Seca

nt (o

f a C

ircle

) A

line

, lin

e se

gmen

t, or

ray

that

pas

ses

thro

ugh

a ci

rcle

at e

xact

ly tw

o po

ints

. The

seg

men

t of t

he s

ecan

t co

nnec

ting

the

poin

ts o

f int

erse

ctio

n is

a c

hord

of t

he c

ircle

. Exa

mpl

e:

se

cant

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

25

Ap

ril 2

014

Sect

or (o

f a C

ircle

) Th

e ar

ea o

r reg

ion

betw

een

an a

rc a

nd tw

o ra

dii a

t eith

er e

nd o

f tha

t arc

. The

two

radi

i div

ide

or s

plit

the

circ

le in

to tw

o se

ctor

s ca

lled

a “m

ajor

sec

tor”

and

a “m

inor

sec

tor.”

The

maj

or s

ecto

r has

a c

entra

l ang

le

of m

ore

than

180

°, w

here

as th

e m

inor

sec

tor h

as a

cen

tral a

ngle

of l

ess

than

180

°. It

is s

hape

d lik

e a

slic

e of

pie

. Exa

mpl

e:

Segm

ent (

of a

Circ

le)

The

area

or r

egio

n be

twee

n an

arc

and

a c

hord

of a

circ

le. I

nfor

mal

ly, t

he a

rea

of a

circ

le “c

ut o

ff” fr

om

the

rest

by

a se

cant

or c

hord

. Exa

mpl

e:

Sem

icirc

le

A h

alf o

f a c

ircle

. A 1

80°

arc.

For

mal

ly, a

n ar

c w

hose

end

poin

ts li

e on

the

diam

eter

of t

he c

ircle

.

Shap

e S

ee fi

gure

.

http://www.mathwords.com/a/arc_circle.htm

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

26

Ap

ril 2

014

Side

O

ne o

f the

line

seg

men

ts w

hich

mak

e a

poly

gon

(e.g

., a

pent

agon

has

five

sid

es).

The

endp

oint

s of

a

side

are

ver

tices

of t

he p

olyg

on.

Sim

ilar F

igur

es

Figu

res

havi

ng th

e sa

me

shap

e, b

ut n

ot n

eces

saril

y th

e sa

me

size

. Ofte

n, o

ne fi

gure

is th

e di

latio

n (“e

nlar

gem

ent”)

of t

he o

ther

. For

mal

ly, t

heir

corre

spon

ding

sid

es a

re in

pro

porti

on a

nd th

eir

corre

spon

ding

ang

les

are

cong

ruen

t. W

hen

sim

ilar f

igur

es a

re n

amed

, the

ir co

rresp

ondi

ng v

ertic

es a

re

liste

d in

the

sam

e or

der (

e.g.

, if t

riang

le A

BC

is s

imila

r to

trian

gle

XYZ,

then

ver

tex

C c

orre

spon

ds to

ve

rtex

Z). E

xam

ple:

Δ

AB

C is

sim

ilar t

o Δ

XYZ

Sine

(of a

n An

gle)

A

trig

onom

etric

ratio

with

in a

righ

t tria

ngle

. The

ratio

is th

e le

ngth

of t

he le

g op

posi

te th

e an

gle

to th

e le

ngth

of t

he h

ypot

enus

e of

the

trian

gle.

sine

of a

n an

gle

= le

ngth

of o

ppos

ite le

gle

ngth

of h

ypot

enus

e

Skew

Lin

es

Two

lines

that

are

not

par

alle

l and

nev

er in

ters

ect.

Ske

w li

nes

do n

ot li

e in

the

sam

e pl

ane.

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

27

Ap

ril 2

014

Sphe

re

A th

ree-

dim

ensi

onal

figu

re o

r sol

id th

at h

as a

ll po

ints

the

sam

e di

stan

ce fr

om th

e ce

nter

. Inf

orm

ally

, a

perfe

ctly

roun

d ba

ll sh

ape.

Any

cro

ss-s

ectio

n of

a s

pher

e is

circ

le. E

xam

ple:

sp

here

Stra

ight

Ang

le

An

angl

e th

at m

easu

res

exac

tly 1

80°.

Surf

ace

Area

Th

e to

tal a

rea

of th

e su

rface

of a

thre

e-di

men

sion

al fi

gure

. In

a po

lyhe

dron

, it i

s th

e su

m o

f the

are

as o

f al

l the

face

s (i.

e., t

wo-

dim

ensi

onal

sur

face

s).

Tang

ent (

of a

Circ

le)

A li

ne, l

ine

segm

ent,

or ra

y th

at to

uche

s a

circ

le a

t exa

ctly

one

poi

nt. I

t is

perp

endi

cula

r to

the

radi

us a

t th

at p

oint

. Exa

mpl

e:

is

a ta

ngen

t of c

ircle

O

http://www.mathwords.com/p/perpendicular.htm

http://www.mathwords.com/r/radius_of_a_circle_or_sphere.htm

K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

28

Ap

ril 2

014

Tang

ent (

of a

n A

ngle

) A

trig

onom

etric

ratio

with

in a

righ

t tria

ngle

. The

ratio

is th

e le

ngth

of t

he le

g op

posi

te th

e an

gle

to th

e le

ngth

of t

he le

g ad

jace

nt to

the

angl

e.

tang

ent o

f an

angl

e =

leng

th o

f opp

osite

leg

leng

th o

f adj

acen

t leg

Tang

ent (

to a

Circ

le)

A p

rope

rty o

f a li

ne, l

ine

segm

ent,

or ra

y in

that

it to

uche

s a

circ

le a

t exa

ctly

one

poi

nt. I

t is

perp

endi

cula

r to

the

radi

us a

t tha

t poi

nt. E

xam

ple:

is

tang

ent t

o ci

rcle

O a

t poi

nt P

Thre

e-D

imen

sion

al F

igur

e A

figu

re th

at h

as th

ree

dim

ensi

ons:

leng

th, w

idth

, and

hei

ght.

Thre

e m

utua

lly p

erpe

ndic

ular

dire

ctio

ns

exis

t.


http://www.mathwords.com/r/radius_of_a_circle_or_sphere.htm


K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

29

Ap

ril 2

014

Tran

sver

sal

A li

ne th

at c

ross

es tw

o or

mor

e lin

es in

ters

ectin

g ea

ch li

ne a

t onl

y on

e po

int t

o fo

rm e

ight

or m

ore

angl

es. T

he li

nes

that

are

cro

ssed

may

or m

ay n

ot b

e pa

ralle

l. E

xam

ple:

lin

e f i

s a

trans

vers

al th

roug

h pa

ralle

l lin

es l

and

m

Trap

ezoi

d A

qua

drila

tera

l with

one

pai

r of p

aral

lel s

ides

, whi

ch a

re c

alle

d th

e ba

ses.

Tria

ngle

A

thre

e-si

ded

poly

gon.

The

mea

sure

s of

its

thre

e in

terio

r ang

les

add

up to

180

°. T

riang

les

can

be

cate

goriz

ed b

y th

eir a

ngle

s, a

s ac

ute,

obt

use,

righ

t, or

equ

iang

ular

; or b

y th

eir s

ides

, as

scal

ene,

is

osce

les,

or e

quila

tera

l. A

poi

nt w

here

two

of th

e th

ree

side

s in

ters

ect i

s ca

lled

a ve

rtex.

The

sym

bol f

or

a tri

angl

e is

Δ (e

.g.,

ΔAB

C is

read

“tria

ngle

ABC

”).

Trig

onom

etric

Rat

io

A ra

tio th

at c

ompa

res

the

leng

ths

of tw

o si

des

of a

righ

t tria

ngle

and

is re

lativ

e to

the

mea

sure

of o

ne o

f th

e an

gles

in th

e tri

angl

e. T

he c

omm

on ra

tios

are

sine

, cos

ine,

and

tang

ent.

Two-

Dim

ensi

onal

Fig

ure

A fi

gure

that

has

onl

y tw

o di

men

sion

s: le

ngth

and

wid

th (n

o he

ight

). Tw

o m

utua

lly p

erpe

ndic

ular

di

rect

ions

exi

st. I

nfor

mal

ly, i

t is

“flat

look

ing.

” The

figu

re h

as a

rea,

but

no

volu

me.





K

eyst

one

Exam

s: G

eom

etry

Asse

ssm

ent A

ncho

r &

Elig

ible

Con

tent

Glo

ssar

y Ap

ril 2

014

Pe

nnsy

lvan

ia D

epar

tmen

t of E

duca

tion

Page

30

Ap

ril 2

014

Vert

ex

A p

oint

whe

re tw

o or

mor

e ra

ys m

eet,

whe

re tw

o si

des

of a

pol

ygon

mee

t, or

whe

re th

ree

(or m

ore)

ed

ges

of a

pol

yhed

ron

mee

t; th

e si

ngle

poi

nt o

r ape

x of

a c

one.

The

plu

ral i

s “v

ertic

es.”

Exa

mpl

es:

Volu

me

The

mea

sure

, in

cubi

c un

its o

r uni

ts3 , o

f the

am

ount

of s

pace

con

tain

ed b

y a

thre

e-di

men

sion

al fi

gure

or

solid

(i.e

., th

e nu

mbe

r of c

ubic

uni

ts it

take

s to

fill

the

figur

e).

Zero

Ang

le

An

angl

e th

at m

easu

res

exac

tly 0

°.

javascript:%20new_definition(link)





Cover photo © Hill Street Studios/Harmik Nazarian/Blend Images/Corbis.

Copyright © 2014 by the Pennsylvania Department of Education. The materials contained in this publication may be

duplicated by Pennsylvania educators for local classroom use. This permission does not extend to the duplication

of materials for commercial use.


Assessment Anchors and Eligible Contentwith Sample Questions and Glossary

April 2014

keystone exams: geometry

Documents