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Keystone Exams: GeometryAssessment Anchors and Eligible Content
with Sample Questions and Glossary
Pennsylvania Department of Education
www.education.state.pa.us
January 2013
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 2
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 Common 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.
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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 Common Core Standard: In the column to the right of each Eligible Content statement is a code representing one or more PA Common 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 Common Core Standards affect this document?
Assessment Anchor and Eligible Content statements are aligned to the PA Common Core Standards; thus, the former enhanced standards are no longer necessary. Within this document, all standard references reflect the PA Common Core Standards.
Standards Aligned System—www.pdesas.org
Pennsylvania Department of Education—www.education.state.pa.us
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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.
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Keystone Exams: Geometry
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.
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MODULE 1—Geometric Properties and Reasoning
Keystone Exams: Geometry
ASSESSMENT ANCHOR
G.1.1 Properties of Circles, Spheres, and Cylinders
Anchor Descriptor Eligible Content
PA Common
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°
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 7
Keystone Exams: Geometry
MODULE 1—Geometric Properties and Reasoning
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°
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 8
MODULE 1—Geometric Properties and Reasoning
Keystone Exams: Geometry
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 9
Keystone Exams: Geometry
MODULE 1—Geometric Properties and Reasoning
ASSESSMENT ANCHOR
G.1.1 Properties of Circles, Spheres, and Cylinders
Sample Exam Questions
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.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 10
MODULE 1—Geometric Properties and Reasoning
Keystone Exams: Geometry
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.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 11
Keystone Exams: Geometry
MODULE 1—Geometric Properties and 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 =
Continued on next page.
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MODULE 1—Geometric Properties and Reasoning
Keystone Exams: Geometry
Continued. Please refer to the previous page for task explanation.
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 =
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 13
Keystone Exams: Geometry
MODULE 1—Geometric Properties and Reasoning
ASSESSMENT ANCHOR
G.1.2 Properties of Polygons and Polyhedra
Anchor Descriptor Eligible Content
PA Common
Core
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.
Sample Exam Questions
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 14
MODULE 1—Geometric Properties and Reasoning
Keystone Exams: Geometry
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 15
Keystone Exams: Geometry
MODULE 1—Geometric Properties and Reasoning
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 16
MODULE 1—Geometric Properties and Reasoning
Keystone Exams: Geometry
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 17
Keystone Exams: Geometry
MODULE 1—Geometric Properties and Reasoning
ASSESSMENT ANCHOR
G.1.2 Properties of Polygons and Polyhedra
Sample Exam Questions
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.
Continued on next page.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 18
MODULE 1—Geometric Properties and Reasoning
Keystone Exams: Geometry
Continued. Please refer to the previous page for task explanation.
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.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 19
Keystone Exams: Geometry
MODULE 1—Geometric Properties and Reasoning
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:
Continued on next page.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 20
MODULE 1—Geometric Properties and Reasoning
Keystone Exams: Geometry
Continued. Please refer to the previous page for task explanation.
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:
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 21
Keystone Exams: Geometry
MODULE 1—Geometric Properties and Reasoning
ASSESSMENT ANCHOR
G.1.3 Congruence, Similarity, and Proofs
Anchor Descriptor Eligible Content
PA Common
Core
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.
Sample Exam Questions
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
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MODULE 1—Geometric Properties and Reasoning
Keystone Exams: Geometry
ASSESSMENT ANCHOR
G.1.3 Congruence, Similarity, and Proofs
Anchor Descriptor Eligible Content
PA Common
Core
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
Continued on next page.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 23
Keystone Exams: Geometry
MODULE 1—Geometric Properties and Reasoning
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 24
MODULE 1—Geometric Properties and Reasoning
Keystone Exams: Geometry
ASSESSMENT ANCHOR
G.1.3 Congruence, Similarity, and Proofs
Sample Exam Questions
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
Continued on next page.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 25
Keystone Exams: Geometry
MODULE 1—Geometric Properties and Reasoning
Continued. Please refer to the previous page for task explanation.
B. What is the length, in units, of } NM ? Show your work. Explain your reasoning.
C. Prove that the measure of /JKL is 90°.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 26
MODULE 1—Geometric Properties and Reasoning
Keystone Exams: Geometry
Standard G.1.3
The diagram shown below is used in a proof.
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:
Continued on next page.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 27
Keystone Exams: Geometry
MODULE 1—Geometric Properties and Reasoning
Continued. Please refer to the previous page for task explanation.
The diagram shown below is used in a proof.
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:
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 28
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
ASSESSMENT ANCHOR
G.2.1 Coordinate Geometry and Right Triangles
Anchor Descriptor Eligible Content
PA Common
Core
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.
Sample Exam Questions
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 29
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
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°)
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 30
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
ASSESSMENT ANCHOR
G.2.1 Coordinate Geometry and Right Triangles
Anchor Descriptor Eligible Content
PA Common
Core
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.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 31
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
Sample Exam Questions
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 32
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 33
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
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
G.2.1 Coordinate Geometry and Right Triangles
Sample Exam Questions
Continued on next page.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 34
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
Continued. Please refer to the previous page for task explanation.
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 35
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
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
Continued on next page.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 36
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
Continued. Please refer to the previous page for task explanation.
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 37
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
ASSESSMENT ANCHOR
G.2.2 Measurements of Two-Dimensional Shapes and Figures
Anchor Descriptor Eligible Content
PA Common
Core
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.
Sample Exam Questions
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 38
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
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°
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 39
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
ASSESSMENT ANCHOR
G.2.2 Measurements of Two-Dimensional Shapes and Figures
Anchor Descriptor Eligible Content
PA Common
Core
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.
Sample Exam Questions
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 40
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 41
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
ASSESSMENT ANCHOR
G.2.2 Measurements of Two-Dimensional Shapes and Figures
Anchor Descriptor Eligible Content
PA Common
Core
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
Sample Exam Question
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.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 42
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
ASSESSMENT ANCHOR
G.2.2 Measurements of Two-Dimensional Shapes and Figures
Anchor Descriptor Eligible Content
PA Common
Core
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
Sample Exam Questions
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 43
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
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%
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 44
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
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
G.2.2 Measurements of Two-Dimensional Shapes and Figures
Sample Exam Questions
Continued on next page.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 45
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
Continued. Please refer to the previous page for task explanation.
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.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 46
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
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:
Continued on next page.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 47
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
Continued. Please refer to the previous page for task explanation.
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:
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 48
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
ASSESSMENT ANCHOR
G.2.3 Measurements of Three-Dimensional Shapes and Figures
Anchor Descriptor Eligible Content
PA Common
Core
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.
Sample Exam Questions
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 49
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
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
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 50
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
ASSESSMENT ANCHOR
G.2.3 Measurements of Three-Dimensional Shapes and Figures
Anchor Descriptor Eligible Content
PA Common
Core
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
Sample Exam Question
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.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 51
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
ASSESSMENT ANCHOR
G.2.3 Measurements of Three-Dimensional Shapes and Figures
Sample Exam Questions
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
Continued on next page.
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 52
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
Continued. Please refer to the previous page for task explanation.
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?
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 53
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
Continued on next page.
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 =
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 54
Keystone Exams: Geometry
MODULE 2—Coordinate Geometry and Measurement
Continued. Please refer to the previous page for task explanation.
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 =
Pennsylvania Department of Education—Assessment Anchors and Eligible Content Page 55
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
Eligible Content Key
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
Eligible Content Key
G.1.3.1.1 C
G.1.3.1.2 D
G.1.3.2.1 C
Eligible Content Key
G.2.1.1.1 D
G.2.1.1.2 D
Eligible Content Key
G.2.1.2.1 B
G.2.1.2.2 A
G.2.1.2.3 A
Eligible Content Key
G.2.2.1.1 C
G.2.2.1.2 A
Eligible Content Key
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
Eligible Content Key
G.2.2.3.1 B
Eligible Content Key
G.2.2.4.1 (page 42) D
G.2.2.4.1 (page 43) D
Eligible Content Key
G.2.3.1.1 D
G.2.3.1.2 A
G.2.3.1.3 B
Eligible Content Key
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
finit
ions
ass
ocia
ted
wit
h th
e Ke
ysto
ne A
sses
smen
t A
ncho
rs a
nd
Elig
ible
Con
tent
. Th
e te
rms
and
defin
itio
ns i
nclu
ded
in t
he g
loss
ary
are
inte
nded
to
assi
st P
enns
ylva
nia
educ
ator
s in
bet
ter
unde
rsta
ndin
g th
e Ke
ysto
ne A
sses
smen
t A
ncho
rs a
nd E
ligib
le C
onte
nt.
The
glos
sary
doe
s 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 pa
rtic
ular
gra
de le
vel o
r co
urse
.
Penn
sylv
ania
Dep
artm
ent o
f Edu
cati
on
ww
w.e
du
cati
on.s
tate
.pa.
us
Janu
ary
2013
K
eyst
one
Exam
s: G
eom
etry
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
2
Janu
ary
2013
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
3
Janu
ary
2013
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
4
Janu
ary
2013
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
5
Janu
ary
2013
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
6
Janu
ary
2013
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
7
Janu
ary
2013
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
8
Janu
ary
2013
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.
K
eyst
one
Exam
s: G
eom
etry
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
9
Janu
ary
2013
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
10
Ja
nuar
y 20
13
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.
K
eyst
one
Exam
s: G
eom
etry
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
11
Ja
nuar
y 20
13
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
12
Ja
nuar
y 20
13
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.
K
eyst
one
Exam
s: G
eom
etry
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
13
Ja
nuar
y 20
13
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
14
Ja
nuar
y 20
13
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.
K
eyst
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Exam
s: G
eom
etry
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
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ary
Jan
uar
y 2
01
3
Pe
nnsy
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epar
tmen
t of E
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tion
Page
15
Ja
nuar
y 20
13
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)
K
eyst
one
Exam
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eom
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Ass
essm
ent
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chor
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t G
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Page
16
Ja
nuar
y 20
13
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.
K
eyst
one
Exam
s: G
eom
etry
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
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ary
Jan
uar
y 2
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ia D
epar
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t of E
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tion
Page
17
Ja
nuar
y 20
13
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
.
K
eyst
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Exam
s: G
eom
etry
Ass
essm
ent
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chor
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ligi
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Con
ten
t G
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ary
Jan
uar
y 2
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Pe
nnsy
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ia D
epar
tmen
t of E
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tion
Page
18
Ja
nuar
y 20
13
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.
K
eyst
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Exam
s: G
eom
etry
Ass
essm
ent
An
chor
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ligi
ble
Con
ten
t G
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ary
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Pe
nnsy
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ia D
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t of E
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tion
Page
19
Ja
nuar
y 20
13
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
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Exam
s: G
eom
etry
Ass
essm
ent
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chor
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ligi
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Con
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t G
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Page
20
Ja
nuar
y 20
13
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:
K
eyst
one
Exam
s: G
eom
etry
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
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ary
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uar
y 2
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ia D
epar
tmen
t of E
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tion
Page
21
Ja
nuar
y 20
13
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
22
Ja
nuar
y 20
13
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
23
Ja
nuar
y 20
13
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°
.
K
eyst
one
Exam
s: G
eom
etry
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
24
Ja
nuar
y 20
13
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
25
Ja
nuar
y 20
13
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
.
K
eyst
one
Exam
s: G
eom
etry
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
26
Ja
nuar
y 20
13
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
27
Ja
nuar
y 20
13
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
K
eyst
one
Exam
s: G
eom
etry
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
28
Ja
nuar
y 20
13
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 th
at it
touc
hes
a ci
rcle
at e
xact
ly o
ne p
oint
. It i
s pe
rpen
dicu
lar t
o th
e ra
dius
at t
hat p
oint
. Exa
mpl
e:
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.
K
eyst
one
Exam
s: G
eom
etry
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
29
Ja
nuar
y 20
13
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
Ass
essm
ent
An
chor
& E
ligi
ble
Con
ten
t G
loss
ary
Jan
uar
y 2
01
3
Pe
nnsy
lvan
ia D
epar
tmen
t of E
duca
tion
Page
30
Ja
nuar
y 20
13
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
°.
Cover photo © Hill Street Studios/Harmik Nazarian/Blend Images/Corbis.
Copyright © 2013 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.
Keystone Exams: Geometry
Assessment Anchors and Eligible Contentwith Sample Questions and Glossary
January 2013