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CALIFORNI.4. ST4.NDARDS TEST
I Geometry
I
THE LOGIC AND GEOMETRIC PROOFS REPORTING CLUSTER
The following seven California content standards are included in the Logic and Geometric Proofs reporting
cluster and are represented in this booklet by 32 test questions. These questions represent only some ways in
which these standards may be assessed on the Geometry California Mathematics Standards Test.
CALIFORNIA CONTENT STANDARDS IN THIS REPORTING CLUSTER
Geometry
GEl .0*
Students demonstrate understanding by identifying and giving examples of
undefined terms, axioms, theorems, and inductive and deductive reasoning.
GE2.0*
Students write geometric proofs, including proofs by contradiction.
GE3.0*
Students construct and judge the validity of a logical argument and give
counterexamples to disprove a statement.
GE4.O*
Students prove basic theorems involving congruence and similarity.
GE5.O
Students prove that triangles are congruent or similar, and they are able to use
the concept of corresponding parts of congruent triangles.
GE6.O
GE7.O*
Students know and are able to use the triangle inequality theorem.
*
Students prove and use theorems involving the properties of parallel lines cut by
a transversal, the properties of quadrilaterals, and the properties of circles.
Denotes key standards
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This is a sample of California Standards Test questions. This is NOT an ope.rational test form, Test scores cannot he projected
based on performance on released test questions. Copyright 0 2009 California Department of Education,
C.4.LIFORr%IA ST\[)RDS TEST
j
Geometry
Which of the following best describes deductive
reasoning?
A
using logic to draw conclusions based on
accepted statements
B
accepting the meaning of a term without
definition
C
defining mathematical terms to correspond
with physical objects
D
inferring a general truth by examining a
number of specific examples
Jj
Consider the arguments below.
1. Every multiple of 4 is even. 376 is a multiple
of 4. Therefore, 376 is even.
II. A number can be written as a repeating
decimal if it is rational. P1 cannot be written
as a repeating decimal. Therefore, p1 is not
rational.
Which one(s), if any, use deductive reasoning?
In the diagram below, Li L4.
A
lonly
B
IT only
C
both I and II
D
neither I nor II
(SG(KJS2
Theorem: A triangle has at most one obtuse
angle.
1
I
Eduardo is proving the theorem above by
contradiction. He began by assuming that in
fABC, LA and LB are both obtuse. Which
theorem will Eduardo use to reach a
contradiction?
4
t
Which of the following conclusions does not
have to be true?
A
Z3 and Z4 are supplementary angles.
B
Line 1 is parallel to line rn.
C
ZlL3
D
Z2L3
A
If two angles of a triangle are equal, the sides
opposite the angles are equal.
B
If two supplementary angles are equal, the
angles each measure 90°.
C
The largest angle in a triangle is opposite the
longest side.
D
The sum of the measures of the angles of a
triangle is 180°.
—6—
a arnp of Cahfoniia Sandaid ioo ue;oos This s NOT an piauonawr form. Tesi scotes cannot tse
based on perfummane on released test stuestions. Copsrtnht
fl)t)9 Califotma Department or Education.
proected
______________________Geometry
(ALIFORNI STANI)ARDS TEST
Use the proof to answer the question below.
Given: ABBC; D is the midpoint of AC
Prove: tABD2CBD
A
D
B
C
Statement
Reason
1.
ABBC; D is the midpoint of AC
1.
Given
2.
ADCD
2.
Definition of Midpoint
3.
BDBD
3.
Reflexive Property
4.
LtABDJLCBD
4.
?
What reason can be used to prove that the triangles are congruent?
A
AAS
R
ASA
Ds.
(S(J!(M
*7*
This is a sample of California Standards Test questions. This is NOT an operational test form. Test scores cannot be projected
based on performance on released test questions. Copyright a) 2009 California L)epartrnent of Education.
CALIFORNJA STANDARDS TEST
Geometry
In the figure below, AB > BC.
C
B
If we assume that in LA = in LC, it follows that
AB = BC. This contradicts the given statement
that AB > BC. What conclusion can be drawn
from this contradiction?
A
,nL4=mZB
B
tiizA,izrB
C
tnL4=,nLC
D
mZAmZC
05000524
—8—
This is a sample of California Standards Test questions, This is NOT an operational test form, Test scores cannot he projected
based on perfOrmance on released test questions. Copyright 9 2009 California Depart.ment of FOucation.
CALIFORNIA STANDARDS TEST
I
Geometry
Use the proof to answer the question below.
Given: L2L3
Prove: L12Z4
£
t
Statement
Reason
1.
Z2L3
1. Given
2.
Z1Z2;L3Z4
2.?
3.
Z1Z4
3. Transitive Property
What reason can be used to justify statement 2?
A
Complements of congruent angles are congruent.
B
Vertical angles are congruent,
C
Supplements of congruent angles are congruent.
D
Corresponding angles are congruent.
CSGIOO9
—9—
This is a .sampie. of California Standards That questions.. This is NOT an operational test form. Test scores cannot be projected
based on performance on released tesr questions. Copyright 02009 California Depart ent of Education,
c:-LJFokNlA STNIHR[)S TEST
Geometry
Eli
“Two lines in a plane always intersect in exactly
one point.”
A conditional statement is shown below.
Which of the following best describes a
If a quadrilateral has perpendicular
diagonals, then it is a rhombus.
counterexample to the assertion above?
A
coplanar lines
B
parallel lines
C
perpendicular lines
D
intersecting lines
Which of the following is a counterexample to
the statement above?
A
C
CS050320
[]
Which figure can serve as a counterexample to
the conjecture below?
f one pair of opposite sides of a
quadrilateral is parallel, then the
quadrilateral is a parallelogram.
B
A
rectangle
B
rhombus
C
square
D
trapezoid
D<
Given: TRAP is an isosceles trapezoid with
diagonals RP and TA. Which of the following
must be true?
A
RPJTA
B
RPJJTA
C
RPTA
D
RP bisects TA
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10
—
This is a sample of California Standards Test questions. This is NOT an operational test form, Test scores cannot he projected
based on erformancc on released test. questions. Copyright © 2009 Cal.ij.omia Department of Education.
CALIFORNIA STANDARDS TEST
I
Geometry
Students in a class rewrote theorems in their
own words. One student wrote the following
statement.
Which triangles must be similar?
The area of a parallelogram is the
product of any base (b) and any height (h).
A
two obtuse triangles
B
two scalene triangles with congruent bases
C
two right triangles
D
two isosceles triangles with congruent
vertex angles
Which figure shows a counterexample to prove
the statement false?
Which of the following facts would be sufficient
to prove that triangles ABC and DBE are
similar?
A
/
b
A
B
b
/
C
b
/E /
E
D
—
11
A
CE and BE are congruent,
B
LACE is a right angle.
C
AC and DE are parallel.
D
LA and LB are congruent.
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This is a srnpie of California Standards That questions. This is NOT ar operatlonal test form. Test scores cannot he projected
based on performance on released t.est questions. Copyright 2009 California Department of Education.
B
C.LIFORNIA STANL)ARDS TEST
I
Geometry
Parallelogram ABCD
is
shown below.
A
In parallelogram FGHI. diagonals IG and FH
are drawn and intersect at point Al. Which of
the following statements must be true?
B
A
AFGI must be an obtuse triangle.
B
AHIG
C
AFMG
D
AGMH must be
must
be an acute triangle.
must
be congruent to AHMG.
congruent to
AIMF.
csosossc
D
C
Which of the following best describes the
triangles shown below?
Which pair of triangles can be established to
be congruent to prove that LDAB ZBCD?
A
AADCandtBCD
B
AAED and LtSBEC
C
ADABandABCD
D
ADECandLtBEA
8
csuss4o
8
If ZABC and AXYZ are two triangles such
that
—
xY
=
—,
Yz
which of the following would
be sufficient to prove the triangles are similar?
A
ZAZX
B
ZBcZY
C
ZCZZ
D
rxzy
A
both similar and congruent
B
similar but
C
congruent
D
neither similar nor congruent
not congruent
but
not similar
050 5,2 15
(SG35
—
12
—
This is a sample of Cali.fornia Standards Test questions. This is NOT an operational test form. Test scores cannot he p.rojected
based on performance on released test questions. Copyright © 2009 California Department of Education.
C3.LIF()RNIA STNI)ARDS TIST
Geometry
Which of the following statements must be true
if 1GHI IJKL?
In the figure below, AC
DF and L4
LD.
C
H
K
AB
G
J
D
A
The two triangles must be scalene.
B
The two triangles must have exactly one acute
angle.
Which additional information would be enough
to prove that /ABC DEF?
C
At least one of the sides of the two triangles
must be parallel.
A
ABDE
B
ABBC
D
The corresponding sides of the two triangles
must be proportional.
C
BCEF
D
BCDE
(SG2K)71
CSGt3O5I7
Which method listed below could not be used to
prove that two triangles are congruent?
A
Prove all three sets of corresponding sides
congruent.
B
Prove all three sets of corresponding angles
congment.
C
Prove that two sides and an included angle of
one triangle are congruent to two sides and an
included angle of the other triangle.
D
Prove that two angles and an included side of
one triangle are congruent to two angles and
an included side of the other triangle.
—
13
*
This is a sampie of California Standards Test questions. This is NOT an operational test fisrrn, Test scores cannot be projected
board on pertormance on released test qucnons. Copyright © 2009 California Depanment of Education.
CL1FORNI
‘
STAN DARDS TEST
I
Geometry
Given: AB and C’D intersect at point E;
Given: E is the midpoint of CD; Z C Z D
Z1L2
A
D
D
B
Which of the following statements must be true?
Which theorem or postulate can be used to
prove /AED1BEC?
A
LALD
A
AA
B
ZBC
B
SSS
C
CEBE
C
ASA
D
ACBD
D
SAS
CSGU)7
CSGI(X)74
E1
In the figure below, n is a whole number. What
is the smallest possible value for a?
n
ii
15
Al
B7
c8
D
—
14
14
—
This is a sample of California Standards Test questions. This is NOT an operational test form. Test scores cannot he projected
based on performance on relrtrard test questions. Copyright © 2009 California Department of Education,
C’.LIFORXIA STANDRDS TEST
Geometry
Which of the following sets of numbers could
represent the lengths of the sides of a triangle?
A
2,2,5
B
3.3.5
C
4.4, 8
D
5,5, 15
What values of a and b make quadrilateral
a parallelogram?
MNOP
In the accompanying diagram, parallel lines 1
and m are cut by transversal t.
A
a=l,b=5
B
a=5,bzl
C
11
34
a=—,b=-—
7
7
D
34
11
a—,b=—
-
rn
N
Which statement about angles 1 and 2 must
be true?
A
Ll2.
B
Zl is the complement of Z2.
C
Zl is the supplement of Z2.
D
/1 and Z2 are right angles.
7
7
Quadrilateral ABCD is a parallelogram. If
adjacent angles are congruent, which statement
must be true?
A
Quadrilateral ABED is a square.
B
Quadrilateral ABED is a rhombus.
C
Quadrilateral ABCD is a rectangle.
D
Quadrilateral ABC’D is an isosceles
trapezoid.
csnoss9
050215)45
—
15
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This is a sample of California Standards Test questions. This is NOT’ an operational test form. Test scores cennot he projected
hased on perfonnance on released test questions. Copnght 02(109 California Department of Education.
CALIFORNIA STANDARDS TEST
I
[Geometry
For the quadrilateral shown below, what is
rnZa + mZc?
The diameter of a circle is 12 meters. If
point P is in the same plane as the circle,
and is 6 meters from the center of the circle,
which best describes the location of point P?
C
A
A
530
B
1370
C
180°
D
233°
A
Point P must be on the circle.
B
Point P must be inside the circle.
C
Point P may be either outside the circle or
on the circle.
D
Point P may be either inside the circle or
on the circle.
0SG00532
E1
Given:
pIIq;
mIIn;
mZ 1=750
If ABCD is a parallelogram, what is the length
of segment BD?
C
What is mZ2?
A
15°
A
10
B
75°
B
11
C
90°
C
12
D
105°
D
14
CSG?(st4t
—
16
—
This is a .sample of California Standards Test questbOns. This .is NOT an operational test form. Test scores cannot he projected
based on perfot.rnan.ce on release.d test questions. Copyright © 2009 California Department of Ed.ucanon,