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8/13/2019 Geometry Standards Workbook
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Name _ Date _
Example 1
California Standards
Geometry 1.0
Students demonstrate understanding by identifying and giving examples
of undefined terms, axioms, theorems, and inductive and deductive
reasoning.
Undefined Terms and Reasoning
Terms to Know Example .d:;;
Undefined terms are words that do In geometry, the words point, line, and
not have formal definitions, but there is plane are undefined terms.
agreement about what they mean.
An axiom, or postulate, is a rule that is Postulate 5:Through any two points, there
accepted without proof. exists exactly one line.
A theorem is a rule that can be proven. Theorem 2:3 (Right Angles Congruence
Theorem): All right angles are congruent.
You use inductive reasoning when you The next number inthe pattern
find a pattern in specific cases and then write 7,14,21, ...a conjecture for the general case.
is 28.
Deductive reasoning uses facts, Sam practices the piano every Tuesday and
definitions, accepted properties, and the laws Thursday.
of logic to form alogical argument. Today is Thursday.
Therefore, Sam practices the piano today.
Identify Undefined TermsWhich ofthe following represents undefined terms?
b. m
d.
Solution
a. A triangle can be described using known words such aspolygon and sides. Itis not
an undefined term.
b.A line is an undefined term.
c.A plane is an undefined term.
d.A ray can be described using known words, such aspoint and line. It is not an
undefined term.
California Standards Review and Practice
Geometry Standards
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Example 2
Example 3
Exercises
Inductive Reasoning
Describe how to sketch the next figure in the pattern. Then sketch the next figure.
Solution
Each figure has one more equal-length side and one more equal-measure angle than
the figure before it.
Answer Sketch the next figure by drawing a figure with six equal-length sides and
six equal-m~asure angles.
Deductive Reasoning
Make a valid conclusion in the situation.
If it rains or snows today, then the Biology field trip will be canceled. It is
raining today.
Solution
Identify the hypothesis and the conclusion of the first statement. The hypothesis is
"If it rains or snows today," and the conclusion is"then the Biology field trip will
be canceled."
"It is raining today" satisfies "the hypothesis of the conditional statement, so you can
conclude that the Biology field trip will be canceled.
Answer The Biology field trip will be canceled.
1. Look for the pattern in the figures shown
below. How many squares will there be in the
tenth figure?
66
45
2 3 4
55
@ 36
2. Ll and L2 are supplementary angles. Which of
the following statements can be justified if
mLI = 110?
L2 is acute because measures ofsupplementary angles have a sum of 90.
L2 isacute because measures ofsupplementary angles have asum of 180.
L2 is obtuse because measures ofsupplementary angles have a sum of 180.
@ L2 isright because m~asures of
supplementary angles have a sum of 180.
California Standards Review and Practice
Geometry Standards 3
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3. The list below shows the volumes of cubes as
the length of the edges is increased. What is the
volume of the eighth cube in the pattern?
1cm3, 3.375 cm3, 8 cm3, 15.625 cm3, ...
48.875 cm3 79.1 cm3
91.125 cm3 512 cm3
4. Which statement about the figures below must
be true?
0600 The four figures are regular.
The four figures are equilateral.
The four figures are equiangular.
The four figures are similar.
5. In isosceles trapezoid ABCD, AB = 28 inches
and DC = 48 inches. What additional data does
not provide sufficient information to find the area
of the trapezoid? .
.the perimeter of the trapezoid
the length ofBC
the measure of LAED
the length ofAE
6. The table shows the dimensions of several
rectangles that fit a pattern. What are thedimensions
of another rectangle thatfits the pattern?
15
Width
45 ,40
50 36
60 30
75 24
90 20
100 18
120
80by42
70by68
72 by 25
65 by 58
California Standards Review and Practice
Geometry Standards
7. Consider the arguments below.
L The number pattern 1,4,9, 16,25,36,49,
64, ... continues forever. The number 800 is
not in the pattern.
II. A quadrilateral's diagonals bisect each
other if itis a parallelogram. A rectangle
is a parallelogram, therefore a rectangle'sdiagonals bisect each other.
Which one(s), if any, use inductive reasoning?
lonly
II only
both Iand II
neither Inor II
8. Look for the pattern inthe dimensions of the
prisms shown below. What will be the volume of
the next figure in the pattern?
L]2 ,1
6
8units3
128 units3
96 units3
256 units3
9. Which of the following does not represent an
undefined term?
m
.x
L'7
8/13/2019 Geometry Standards Workbook
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California Standards
Geometry 2.0
Students write geometric proofs, including proofs by
contradiction.
Geometric Proofs
A proof is a logical argument that shows a statement is true.
Reason logically from the given information, making one statement at
a time, until you reach the conclusion.
STEP 1 Identify the given information and the statement you want to prove.
STEP 2
STEP 1 Identify the given information and the statement that you want to
prove. Assume that this statement is false by assuming that its
opposite istrue.
STEP 2 . Reason logically, making one statement at a time, until you reach
a contradiction.
STEP 3 State that the desired conclusion must be true because the
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Write a Direct Proof
GIVEN ~ AC = BD
PROVE ~ AB = CD A B c D
Solution
Statements Reasons
1.AC = BD
2.AB + BC=AC
3.BD =BC+ CD
4.AC= BC+ CD
5. AB + BC = BC + CD
6.AB = CD
1. Given
2. Segment Addition Postulate
3. Segment Addition Postulate
4. Substitute AC for BD.
5.Transitive Property of Equality
6. Subtract BC from both sides.
California Standards Review and Practice
Geometry Standards 5
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Example 2 Write a Proof by Contradiction
AGIVEN ~ mL.A = 115
PROVE ~ L. B is not a right angle.
B
cSolution
STEP 1 Assume that L.B is a right angle.
STEP 2 If L.B is a right angle, then the sum ofthe measures ofthe other two angles
in the triangle must be 90: m L.A + m L.C = 90. Therefore m L.A =
90 - m L. C,so m L.A -c
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3. In the figure below, L 1 'i=L2 and DE == EF.
E
&o x F
If we assume that DX == XF, and use EX == EX by
the Reflexive Property of Segment Congruence,
then 6.DEX == 6.FEXby SSS. Wecan conclude
that L 1 == L 2 because corresponding parts
of congruent triangles are congruent. This
contradicts the given statement that L 1 'i=L 2.
What conclusion can be drawn from this
contradiction?
Ll ==L2
Ll'i=Ll
DX'i=XF
DE'i=EF
4. Given: s $ t
Prove: Lines sand t intersect at exactly one point.
-s
t
Consider the two assumptions.
1. Lines sand t intersect at more than one point.
II. Lines sand tdo not intersect.
Which one(s), if any, would you use to write a
proof by contradiction?
Ionly
lIonly
both I and II
neither Inor II
5. Use the proof to answer the question below.
Given: AB == CD, CD == EF
Prove: AB == EF
Bc
- o
-
Statement Reason
1.GivenLAB == CD;-- -
CD==EF
2.AB = CD;CD=EF
3.AB = EF
- -
4.AB == EF
2. Definition ofcongruent segments
3. ?
4. Definition of
congruent segments
Which reason can be used to justify Statement 3?
Symmetric Property
Transitive Property
" Reflexive Property
Ruler Postulate
6. Susan wants to prove that the hypotenuse
of a right triangle is the longest side. What
assumption should she make to write a proof
by contradiction?
p
R' , ........Q
PR +RQ>PQ
PR+RQ
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California Standards
Geometry 3.0
Students construct and judge the validity of a logical argument.
and give counterexamples to disprove a statement.
Conditional Statements and Counterexamples
A conditional statement is a logical statement made up of a hypothesis and
a conclusion. Itis often written in if-then form:
If two angles are both right angles, then they are congruent.
i ihypothesis conclusion
Conditional statements can be either true or false. If you want to show that a
conditional statem~nt is true, then you must prove that the conclusion is true whenever
the hypothesis is true. If you want to show that a conditional statement is false, youneed to give only one counterexample. A counterexample is a specific cas.e for
which the hypothesis is true but the conclusion is false.
By rearranging or negating the hypothesis and conclusion of a conditional statement,
you can form related conditionals.
" '" 'k,.True or
Related ConJlitional Examplefalse?
Counterexample, . ,,/i
Conditional If two angles are True
-
statement both right angles,
then they are
congruent.
Converse If two angles are False
Q~LExchange the hypothesis congruent, then
and the conclusion. they are both
right angles.
y z
Inverse If two angles False
Q~LNegate both the hypothesis are not both
and the conclusion. tight angles,
then they are not
congruent.y z
Contrapositive If two angles are True
Write the converse, then' not congruent,
negate both the hypothesis then they are not
and the conclusion. both right angles.
A conditional statement and its contrapositive are either both true or both false. Also,
the converse and the inverse of a conditional statement are either both true or both false.
California Standards Review and Practice
Geometry Standards
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Example 1
Example 2
Analyze Conditional Statements
Write the if-then form, the converse, the inverse, and the contrapositive of the
conditional statement. Decide whether each statement is true orfalse.
Parallelograms are quadrilaterals.
Solution
If-then form: Ifafigure is a parallelogram, then itis a quadrilateral.A parallelogram is a quadrilateral. The statement is true.
Exchange the hypothesis and the conclusion.
Ifafigure is a quadrilateral, then it is a parallelogram.
Counterexample: A trapezoid is a quadrilateral, but it is not a
parallelogram. The statement is false.
Negate both the hypothesis and the conclusion.
Ifafigure is not a parallelogram, then it is not a quadrilateral.
Counterexample: A trapezoid is not a parallelogram, but it is a
quadrilateral. The statement is false.
Contrapositive: . Write the converse.Ifafigure is a quadrilateral, then it is a parallelogram.Negate both the hypothesis and the conclusion.
Ifafigure is not a quadrilateral, then itis not a parallelogram.
Converse:
Inverse:
If a figure does not have four sides, it can't be a parallelogram. The
statement is true.
Find Counterexamples
Show that the conjecture is false by finding a counterexample.
a. IfJK = KL, then Kis the midpoint ofJL.
b. IfAB = BC = CD = DA, then quadrilateral ABCD is a square.
Solution
a.J, K, and Ldo not have to be collinear.
K
b. Quadrilateral ABCD may not have a right angle.
, B, } I
o c
California Standards Review arid Practice
Geometry Standa'rds 9
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Exercises
Identify the statement that has the same
meaning as the given statement.
1. The seafood restaurant is closed every Monday.
Ifthe seafood restaurant is closed, then
itis Monday.
If.it is Monday, then the seafood restaurant
is not closed.
If it isMonday, then the seafood restaurantis closed.
If it is not Monday, then the seafoodrestaurant isnot closed.
2. You can buy a new CD once you have saved
enough money.
If you have saved enough money, then you, can buy a new CD.
If you can't buy a new CD, then you havesaved enollgh money.
If you have not saved enough money, thenyou can buy a new CD.
If you buy a new CD, then you have notsaved enough money.
3. You are told that a conditional statement is false.
Consider the related conditionals.
I. Inverse
II. Contrapositive
III. Converse
Which one(s) is (are) also false?
Ionly
III only
II only
both I and III
4. "Through any three points there exists exactly
one plane."
Which ofthe following best describes a
counterexample to the conjecture above?
parallel planes
" perpendicular lines
collinear points
parallel lines
California Standards Review and Practice
0 Ge-emetry Standards
5. IfDEFG is a parallelogram with diagonals DF
and EG, which of the following must be true?
DF=EG
DE=DG
DF bisects EG.
DEl.DG
6. A conditional statement is shown below..
If L I and L2 are complementary,
then they form ar(ght angle.
Which of the following is a counterexample to
the statement?
2
2
7. Which statement is sufficient to prove that L 1
and L2 are complementary?
Ll and L 6 are supplementary.
L 2 and L 4 are complementary.
L 1 and L 7 are supplementary.
- L 5 and L 8 are complementary.
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Reflexive
Segment and angle congruence are reflexive, symmetric, and transitive .
For any segment AB, AB :::= AB. For any angle A, LA :::= LA.
Symmetric IfAB :::= CD, then CD :::= AB. If LA :::= LB, then LB :::= LA.
Transitive IfAB :::= CD and CD :::= EF, then AB :::= EF. If LA :::= LB and
LB:::= LC, then LA:::= LC.
California Standards Review and Practice
Geometry Standards 11
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Example 1
Example 2
Congruent Complements Theorem
If two angles are complementary to the same angle (or to congruent angles), then they are
congruent.
If L 4 and L 5 are complementary and L 6 and L 5 are complementary, then L 4 :=L 6.
Vertical Angles Congruence Theorem
Vertical angles are congruent.
LI:=L3,L2:=L4
Congruence
GIVEN ~ PR bisects L QPS.PS bisects LRPT.
PROVE ~ L QPR := L SPT
p
Solution
Statements Reasons
1.PRbisects L QPS.
PS bisects L RPT.
2. LQPR:= LRPS
3. LRPS:= LSPT
4. LQPR:= LSPT
1.Given
2. Dc:finition of Angle Bisector
3. Definition of Angle Bisector
4.Transitive Property of Angle Congruence
Similarity
In the diagram, IiPQR ~ Ii STU. Find the value ofx.
Solution
The triangles are similar, so the corresponding side
lengths are proportional.
p
12~PR SU
PQ = ST
.18 1221 - x
Write.a proportion.R 15 Q U 10 T
Substitute.
18x = 252
x =14
Cross Products Property
Solve for x.
California Standards Review and Practice
Geometry Standards
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Exercises
1. De{ermine which pair of triangles is similar.
@ two scalene triangles
@ two isosceles triangles
two right triangles
two equilateral trian~les
2. In the diagram, LABC ~ LE DF. What is the
value ofy?
A
D
B~
F E
16
C
@6 9 @ 12@8
3. In the figure below, mL CFD = 90.
A D
Which pair of angles cannot be proven
congruent?
@ LBFC,LEFD
LCFD,LCFA
@ LAFB,LEFD
@ LAFE,LBFD
4. Which statement about the figure is not true?
71@ L3 and L6 are supplementary.
@ L 1 and L 5 are supplementary.
L3 and L4 are complementary.
@ L2 and L 5 are complementary.
.5. Inthe figure, LPQR ~ LXYZ. Which statement
must be true?
p
X
Q [>YzR
@ The two triangles are isosceles.
@ The two triangles are congruent.
The corresponding sides of the two trianglesare congruent.
@ The corresponding angles of the two
triangles are congruent.
6. Use the proof to answer the question below.
Given: L WVY and L XVZ are right angles.
Prove: L YVZ == L WVX
w
v.....,
Statement
1.L WVYand LXVZ
are right angles.
2. L WVH and LXVY
are complementary.
Reason
1. Given
2. Definition of
complementary angles
3.Definition of
complementary angles
4. ?
3. LWVYand LYVZ
are complementary.
4. LYVY== LWVX
Which reason can be used to justify Statement 4?
@ Vertical Angles Congruence Theorem
@ Symmetric Property of Congruent Angles
Congruent Complements Theorem
@ Congruent Supplements Theorem
California Standards Review and Practice
Geometry Standards 13
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California Standards
Geometry 5.0
Students prove that triangles are congruent or similar, and they are able
to use the concept of corresponding parts of congruent triangles.
Triangle Congruence and Similarity
In two congruent figures, all the parts of one figure are congruent to the
corresponding parts ofthe other figure.
Corresponding angles:
LA :=LF,LB:= LE,LC:= LD
Corresponding sides:
AB :=FE, BC:=ED,AC:=FD
B E
~~A C F 0
6ABC:= 6FEDWhen you write a congruence statement for two polygons,
always list the corresponding vertices in the same order.
Side-Side-Side (SSS) Congruence
Postulate
If three sides of one triangle are congruent to
three sides of a second triangle, then the two
triangles are congruent. 6ABC:= 6PQR
Side-Angle-Side (SAS) Congruence
Postulate
If two sides and the included angle of one
.triangle are congruent to two sides and the
included angle of a second triangle, then the
two triangles are congruent. 6DEF:= 6STU
Angle-Side-Angle (ASA) Congruence
Postulate
If two angles and the included side of one
triangle are congruent to two angles and the
included side of a second triangle, then the two
triangles are congruent. 6DEF:= 6MNO
B Q
66A CPR
L1~o F S U
E. N
D~ M~F o
Hypotenuse-Leg (HL) Congruence
Theorem
If the hypotenuse and a leg of a right triangle
are congruent to the hypotenuse and a leg of asecond right triangle, then the two triangles are
congruent. 6JKL:= 6XYZ I K
J X
~~L y z
Angle-Angle-Side (AAS) Congruence
Theorem
H W
If two angles and a non-included side of one I /\ /\ triangle are congruent to two angles and a non- ~ ~
included side of a second triangle, then t4e two I G
triangles are congruent. 6 GHI:= 6 VWX
v X
California Standards Review and Practice
Geometry Standards .
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Example 2
Example 3
Determine Information to Show Congruence
State the third congruence that must be given to prove that LABC == LPQR using the
indicated postulate or theorem. B Q
a. LB == LQ, LC == LR
Use the ASA Congruence Postulate.
b.BC ==
QR, LB ==
LQUse the SAS Congruence Postulate.
Solution c R
a. Two angles in the first triangle are congruent to two angles in the second triangle. To
use the ASA Congruence Postulate, we need to know that the included side in the
first triangle is congruent to the included side in the second triangle, or BC == QR.
c. One side and one angle in the first triangle are congruent to one side and one angle
in the second triangle. To use the SAS Congruence Postulate, we need to know that
another side of the first triangle is congruent to the corresponding side of the second
triangle, such that the congruent angles are the included angles. So,AB == PQ.
Determine Whether Triangles Are Congruent
Decide whether the congruence statement is true. Explain your reasoning.
a. LWYZ== LYWX wA .:;> X
.b. L VXY == Lzxw
z
V
y
X
z
c. LJKL == LMNO K N
J M
L .0
Solutiona.Yes, by the HL Congruence Theorem. L WYZ is a right angle by the Corresponding
. Angles Postulate. WY == WYby the Reflexive Property of Congrent Segments, and
ZW == XY is given.
b.Yes,by the AAS Congruence Theorem. LX== LXby the Reflexive Property of
Congruent Angles and L V == L Z is given. ZW = ZT+ TWand VY = VT+ TYby
the Segment Addition Postulate. ZW = VYby the Transitive Property of Equality,
and ZW == VYby the Definition of Congruent Segments.
c. No; SSA isnot one of the triangle congruence postulates or theorems.
California Standards Review and Practice
Geometry Standards'6
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Example 4
Example 5
Example.6
Date _
Use Corresponding Parts
Write a congruence statement for the triangles. Identify an pairs of congruent
corresponding parts.
p
Solution
The diagram indicates that 6.ABC == 6.RPQ.
Corresponding angles LA == LR, LB == LP, L C == L Q
Corresponding sides AB == RP, BC == PQ, CA == QR
ShowTriangles Are Similar
Show that the triangles are similar and write a similarity statement. Explain your
reasomng.
P 3
~RT 4 S 16
SQlution
Since we know the lengths of the sides, calculate the ratios of corresponding sides.
QS 8 4 QR 12 12 4 SR 16 16 4
PT - 10 - '5 PR 3 + 12 - 15- '5 TR - 4 + 16 - 20 - '5
;: = ;; = ~~.= ~, thus the triangles are similar by the SSS Similarity Theorem.
Answer 6.TPR ~ 6.SQR by the SSS Similarity Theorem.
Prove Triangles Are Similar
GIVEN ~ KP==LP,JL = 21,KM= 14,
LQ = 24,NK= 16
J K L M<
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Exercises
Date _
1. /:"IKL and /:"PQR are two triangles such that
L K := L Q.Which of the following is sufficient
to prove the triangles are similar?
@ JK=PQ
LJis right.
2. In the figure below, wzI I XY.
V
I ". Z
. X y
Which theorem or postulate can be used to prove
/:,. VWZ~ /:"VXY?
@ ASA AAS SAS SSS
3. In the figure below, /:,.ABE:= !iDCE.
Ai"":"
B
oIL -c
Which theorem or postulate can be used to prove
/:"CDB:= /:"BAC?
@ ASA SSS SAS AAS
4. In the figure below, PQ IISR.
I AQ
sv I I
Which additional information would be enough
to prove /:"PQS:= /:,.RSQ?
@ PQ:=PS
PQ:=SR
SR:=QR
PS:=QR
California Standards Review and Practice
8 Geometry Standards
5. In the figure below, HI bisects LKHI and LKII.
H
K
J
Which theorem or postulate can be used to prove
/:"HKI:= /:"HIJ?
@ ASA
SAS
AAS
SSS
6. In the figure below, L P := LX .
X
P
R~
Z
Q y
Which of the following would be sufficient to
prove the triangles are similar?
RPZX
RP
ZX
PQXY
RQ
ZY
7. In the, figure below, ED ..1DF, HG..l GF, F is
the midpoint ofDG.
E
H
Which theorem or postulate can be used to prove
/:"DEF:= /:,.GHF?
@ ASA SSS SAS HL
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Example 2
Example 3
Example 4
Use the Triangle Inequality Theorem
A triangle has one side of length 17 and another of length 11.Describe the possible
lengths ofthe third side.
Solution
Let x represent the length of the third side. Draw diagrams to help you visualize the
possible lengths ofthe third side. "
Small values of x Large values of x
17 x
~ ~
x+ll>17
x> 6
11+17>x
28 >x, or x < 28
Answer The length of the third side must be greater than 6 and less than 28.
Use the Triangle Inequality Theorem
Describe the possible values of x.
7x-17
Solution
Check all three possible side length relationships.
(x + 6) + (2x + 7) >7x - 17
3x+ 13>7x-17
30>4x1
7->x2
13 x + 6 (x + 6) + (7x - 17) >2x + 7
9x - 10 > x + 6 8x - 11 > 2x + 7
8x>16 6x>18
x>2 x>3
Answer
Use the Triangle Inequality Theorem
The triangle below is isosceles. If s is a whole number, what is its smallest possible value?
23
Solution
Use the Triangle Inequality Theorem to write and solve an inequality.
s + s > 23
2s> 23
1s> 112
The smallest whole number that is greater"than 11k is 12.
Answer The smallest possible value for sis 12.
California Standards Review and Practice
0 Geometry Standards
8/13/2019 Geometry Standards Workbook
20/81
Name _
Exercises
Date _
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1. Two sides ofa triangle measure 14 and 9.Which.
of the following cannot be the perimeter of the
triangle?
@ 28
42
37
46
2. The lengths of two sides of the triangle are
known.
7
Which of the following could be the perimeter of
the triangle?
@ 19
31
24
38
3. The figure shows the route Daniel took while
riding his bicycle after schooL
5 mi
Which of the following is not apossible measure
for the third side of the triangle?
@ 4mi
6mi
5mi
7mi
4. The figure shows the outline of a nower garden.
Which of the following is a possible measure for
the third side of the garden?
@ 4ft
20ft
8ft
24ft
5. The triangle below is isosceles.
If tis a whole number, what is its largest possible
value?
@ 35 36 37 38
6. A triangle has one side of length 12 and another
of length 19. Which of the following best
describes the possible lengths of the third side?
@ 7
8/13/2019 Geometry Standards Workbook
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Name ~ __
California Standards
Geometry 7.0
Students prove and use theorems involving the properties
of parallel lines cut by a transversal, the properties of
quadrilaterals, and the properties of circles.
Date _
Parallel Lines, Quadrilaterals, and Circles
Pa~anel'Llhe; andTransver~al$,
L2=L6
m
n
Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then the
pairs of corresponding angles are congruent.
The converse is also true,
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the
pairs of alternate interior angles are congruent.
The converse is also true.
1.
L4= L5
m
n
Alternate Exterior Angles TheoremIf two parallel lines are cut by a transversal, then the
pairs of alternate exterior angles are congruent.
The converse is also true.
L1 = L8
m
n
Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs of consecutive interior angles are
supple~entary.
The converse is also true.
1.
L 3 and L5 are supplementary.
.m
n
California Standards Review and Practice
2 Geometry Standards
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Name ,Date
Example
If a quadrilateral is a parallelogram, then its opposite Q, II ,R
sides are congruent.
The converse is also true.
p
If a quadrilateral is a parallelogram, then its opposite Q, } \I,R
angles are congruent.
The converse is also true.
p
If a quadrilateral is a parallelogram, then itsQj Xo jR
consecutive angles are supplementary.
IfPQRS is a parallelogram, then XO +yO = 180.1
/ Vo Xo
P
If a quadrilateral is a parallelogram, then its QK
>IRdiagonals bisect each other.
The converse is also true.
p
If one pair of opposite sides of a quadrilateral are Q, I
,R
congruent and parallel, then the quadrilateral is a
parallelogram.
Ip
$pe~ial;c
:)i;>.,,}
Example ..-
Parall~to9rams-"
A quadrilateral is AUea rhombus if andonly if it has fourcongruent sides.
A quadrilateral is a
:0:rectangle if and onlyif it has four rightangles.,
A quadrilateral is a ADBsquare if and only ifit is a rhombus and a
re
8/13/2019 Geometry Standards Workbook
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8/13/2019 Geometry Standards Workbook
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Example 1
Example 2
Example 3
.Parallel Lines and Transversals
Find mL 1 and mL 2. Explain your reasoning.
Solution
L2 and the given angle are alternate exterior
angles. So, L2 is congruent to the given angle
by the Alternate Exterior Angles Theorem.
Therefore, mL2 = 110.
Since L 1 and L2 form alinear pair, they are
supplementary angles. So,
mL 1 = 180 - mL2 = 180 - 110 = 70.
Prove That Figures Are Congruent
Write a proof.
GIVEN ~ kite FGHI with diagonal GI
PROVE ~ 6.IFG == 6.IHG
Solution
Because FGHI is a kite"FG == HG and FI == HI.
By the Reflexive Property, GI == GI.
So, 6.IFG == 6.IHG by the SSS Congruence Postulate.
F
Find Measures of Arcs
AC is a diameter ofcircle E. Identify the given arc as a major arc, minor arc, orsemicircle, and find the measure of the arc.
~a.AD
c.ABC
~b.DBC
d.BC
Solution
a. mAD = mADe - mOC b.mDBC = 360 - mOC
= 180 - 80
= 100
mAD is less than 180. Itis a minor arc.
c.AC is a ,diameter.
= 360 - 80
= 280
~
mDBC ismore than 180. Itis amajor arc.
d.mBC =mABC - mAE~
mABC is 180. Itis semicircle. = 180 - 45
= 135.~
mBC is less than 180. It is a minor arc.
California Standards Review a nd Practice
Geometry Standards 25
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Name ~ ~-------- Dat~ _
Exercises
What is the length ofPQ?
'W Z
I
004 4Y2
003 6 8 8Y2
8 10
1. Identify the postulate or theorem that justifies the
statement aboutthe diagram.
L2=L7
00 Corresponding Angles Postulate
Alternate Exterior Angles Theorem
Alternate Interior Angles Theorem
Consecutive Interior Angles Theorem
2. What is the value ofx in the diagram?
2x
3. Quadrilateral WXYZ is a trapezoid. XY = 6 and
WZ= 10. What is MN?
California Standards Review and Practice
6 Geometry Standards
4. Quadrilateral lKLM is a parallelogram. Ifits
diagonals are perpendicular, which statement
must be true?
00 Quadrilateral JKLM isarectangle.
Quadrilateral lKLM isa rhombus.
Quadrilateral JKLM isan isosceles
trapezoid.
Quadrilateral JKLM is a square.
5. GHand lK are diameters ofcircle C. If~ .------....
mHK = 35, what is mG1K?
6. In the figure below, PQRS is a parallelogram.
r Q I \
(4a+b)"
1100
p
What are the values ofaand b?
00 a = 70, b = 110 a = 110, b = 70
a = 30, b = 20 a = 20, b = 30
7. In the figure below, circle M has adiameter of 8~
and mPQ = 90.
-,( >P
>-cco0.
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Name _ Date __
California Standards
Geometry 8.0
Students know, derive, and solve problems involving perimeter,
circumference, area, volume, lateral area, and surface area of
cominon geometric figures.
Perimeter, Area, andVolume
The perimeter
0of a figure is the 0, Adistance around it.
s w f---b----j
P = 4s P = 2 + 2w P=a+b+c
Circumference
Eijis the distancearound acircle.C = 1Td= 21Tr
Area is the amount
0, A Eijof surface covered by a figure.>- w f---b----jcco0..
A = 1bhE A =w A = 1T?aU 2
c
~The volume of a~
c solid is the number 1 I I Ih.s I.s:::
of cubic unitsOl:::J
a
contained in its:::r::.....a interior.
Iwc
a
'CijV= Bh = whs
'6co
A face of a solidCD
that is not a base is I. fjOlI\!$ 'lill~Et~r;:~h:::0~co
,a lateral face.
Ol
The lateral area:::Ja0
of a solid isthe sumu2>- of the areas of its I L = 2M + 2hw.09 lateral faces.....,.s:::
OJ
.~ I The surface0..
a area of asolid is ~~~~I
8/13/2019 Geometry Standards Workbook
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Name ~ _
Example 1
Example 2
Example 3
Example 4
Find the Unknown Length
Date _
The perimeter of the triangle is 17.25 feet. Find the length ofb.
Solution
P=a+b+c
17.25 = 5 + 6.5 + b
17.25 = 11.5 + b
5.75 = b
Answer The length ofb is 5.75 feet.
Find the Circumference
b
A circular stained-glass window has a diameter of 80 centimeters. Find the
approximate circumference of the window. Use 3.14 for 7r.
Solution
C = 7rd
C = 3.l4(80) = 251.2
Answer The circumference ofthe window is approximately 251.2 centimeters.
Find the Area
In the diagram, the diameter of the large circle is three times the diameter of the small
circle. What fraction of the large circle is covered by the shaded region?
Solution
Small circle:
Large circle: A = 7r?- = 7r(3x)2 = 97rx2
Shaded region: A = 97rx2 - 7rX 2 = 87r~
Area of shaded region 87TX2 _ 8
Area oflarge circle - 9 7T~ 9
Answer The shaded region 'covers ~ of the large circle.
Find the Lateral Area
The lateral area of the cylinder is 376.8 square inches. Find the height of the cylinder. Use 3.14
for 7r. (The lateral area of a right cylinder is27rrh,where h is the height of the cylinder.)
Solution --
L = 27rrh
376.8 = 2(3.l4)(4)h
376.8 = (25.12)h
h = 15
Answer The height of the cylinder is 15 inches.
California Standards Review and Practice
Geometry Standards
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Name ___ Date _
5. Justine ispainting rectangular panels in a
restaurant. She isusing a can of enamel that
covers at most 200 square feet. She has painted a
panel that is S feet by S feet and a second panel
that is 3 feet by 10 feet. She just manages to
paint one of the remaining four panels before
she has to open another can. What are the most
reasonable dimensions of the third panel?
@ lOft by 13 ft
lOftbySft
10 ftby 10 ft
9ftbySft
6. A company makes a cylindrical cardboard
container with the dimensions shown below.
20 in.
What isthe approximate lateral area?
@ 1257 in.2
5027 in.2
2513 in.2
12,566 in.2
7. An architect designed a window with the
dimensions shown below.
T21 in.
. .1f--28 in.---j
What is the area of the window to the nearest
square inch? Use~3.14 for 7T.
@ 3050in.2
1203 in.2
1S19 in.2
S96in.2
California Standards Review and Practice
Geometry Standards
8. Ava made a pencil holder in the shape of an
open square prism that has a volume of 96 cubic
inches. If the sides of the base are 4 inches long,
what is the height?
@ 4 in.
Sin.
6 in.
24 in.
9. Joshua has tied his horse's rope to a post in a
pasture so that the horse can eat some grass.
The portion of the rope between the horse and
the post is 12 feet long. To the nearest whole
number, what is the area of the circular region
of the pasture where the horse will be able to
graze?
@ 452 ft2
75 ft2
10. Eli has roped off a square inside his circular pool
so that he and his friends can playa game.
To the nearest tenth, what isthe area of the pool's
surface that is not contained within the roped-off
square region?
@ 1S.S yd2
10.3yd2 14.3 yd
2
9.0yd2
11. What is the volume of this solid?
@ l4yd3
169yd3
51 yd3
270yd3
--
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California Standards
Geometry 9.0
Date _
Students compute the volumes and surface areas of prisms,
pyramids, cylinders, cones, and spheres; and students commit
to memory the formulas for prisms, pyramids, and cylinders.
Volume and Surface Area
A prism is a polyhedron with two bases
that are congruent polygons in parallel
planes.
In a right prism, each lateral edge is
perpendicular to both bases.
A prism with lateral edges that are
not perpendicular to the bases is an
oblique prism.
triangularprism
A pyramid is a polyhedron with a
polygon for its base. All of the other
faces intersect at one vertex.
A regular py.ramid has a regular
polygon for a base, and the segment
joining the vertex and the center of the
base is perpendicular to the base. triangular pyramid
S-apothem
pentagonalprism
A cylinder is a solid with congruent
circular bases that lie in parallel planes.
In a right cylinder, the segment
joining the centers of the bases is
perpendicular to the bases.
In an oblique cylinder, the segment
joining the centers of the bases is not
perpendicular to the bases.
cylinder
heig~t slant, height,,..---,
,B
rectangular pyramid
right cylinder
A cone has a circular base and avertex
that is not in the same plane as the base.
In a right cone, the segment joining
the vertex and the center of the base is
perpendicular to the base.
/~G
cone
he;g~nt he;ght
,- -- \B ;
right cone
A sphere is the set ofall points in
space equIdistant from a given point,
called the center.
sphere
California Standards Review and Practice
Geometry Standards 31
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Name_ __
Example 1
Example 2
Date _
. 1. y ,i;; "'1\1' ' /' "" ,.~ !il ,; ..'Solid., Volume' " .Surface 1:4iea
~:, '''... . '", c', ;~,~ '''' '< '{ :;",~ .'" ' ( f
prism
B is the area ofa base, h is the height,V=Bh S = 2B + Ph = aP + Ph
a isthe apothem ofa base, and P isthe
perimeter ofa base.- (right prism)
pyramid
B is the area of the base, h is theV= lBh S = B + ~P'
height, P is the perimeter of the base, 3
and . isthe slant height. (regular pyramid)
cylinder
B is the area of a base, h is the height,
r is the radius of a base, C is the V= Bh = 7T?h S = 2B + Ch = 27T? + 27Trh
circuinference of a base, and h is the
height.
cone
B is the area of the base, h is theV= lBh = 17T?h S = B + ~ CL = 7T? + 7Tr.height, r is the radius ofthe base, Cis
3 3'the circumference of the base, and . is (right cone)the slant height.
sphereV= ~7T,3 s = 47T?'
r is the radius. 3
Find the Volume of a Prism
Find the volume of the right prism.
Solution
Find the ~rea ofthe base. B = ~h(bi + b2) = ~(6)(4 + 11) = 45
[Note: h in this area formula is the height of the trapezoid, not the
height ofthe prism.]
11 ft
~2ft
4ft
Find the volume. V = Bh = (45)(2) = 90
Answer. The volume of the prism is 90 ft3~
Find the Volume of a Pyramid
Find the volume of the pyramid.
Solution
Find the area of the base. B = ~bh = ~(9)(7) = 31.5
[Note: hin this area formula is the height of the triangle, not the
height ofthe pyramid.]
Find the volume. V= i Bh = i(31.5)(8) = 84
Answer The volume of the pyramid is 84m3.
California Standards Review and Practice
Geometry Standards
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Example 3
Example 4
Example 5
Example 6
Find the Volume of a Sphere
Find the volume of the sphere. Use 3.14 for 1T.
Solution
The diameter is given. Find the radius. r = ~ = 124 = 7
V = ~ 1Tr3 = ~ (3.14)(73) = 1436
Answer The volume of the sphere is approximately 1436 cm3.
Find the Surface Area of a Cylinder
Find the surface area of the cylinder. Use 3.14 for 1T.
Solution
The diameter is given. Find the radius. r = ~ = ~o= 20
Find the surface area.
s = 21Tr'2 + 21Trh = 2(3.14)(202) + 2(3.14)(20)(16) = 4522
Answer The surface area of the cylinder is approximately 4522 ft2.
Find the Surface Area of a Cone
4 in.Find the surface area ofthe cone. Use 3.14 for 1T.
Solution
Find the slant height. J!2= 42 + 32
J!=V!6+9=V25=5
S = 1Tr'2 + 1TrJ! = 3.14(32) + 3.14(3)(5) =75Find the surface area.
Answer The surface area of the cone is approximately 75 in.2.
Find the Surface Area of a Sphere
3 in.
3
Find the surface area of the sphere. Use 3.14 for 1T.
Solution
S = 41Tr'2 = 4(3.14)(122) = 1809
Answer The surface area of the sphere is approximately
1809 mm2.
16ft
California Standards Rev.iew and Practice
Geometry Standards 33
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Name ~-------- Date _
Exercises
1. Jose wants to calculate the volume of air in a
building, shown below, so that he can decide on
the size of a new furnace. What is the volume of
the building?
@ 8280 ft3
72,000ft3
82,800 ft3
93,600 ft3
2. A triangular prism is shown below. Itsvolume
is 672 cubic inches. What is {he height xof the
prism?
@ 7 in.
12 in.
lOin.
17.l in.
3. The diagram represents a sculpture in an art
museum. What is the surface area of the sculpture?
Round your answer to the nearest tenth. Use 3.l4
for 1T.
@ 113.8 ft2
241.7 ft2
1-3 fH
127.9 ft2
30l.8 ft2
California Standards Review and Practice
Geometry Standards
4. Which of the following is the approximate
volume of a ball that has a diameter of 9 inches?
Use 3.14 for 1T.
@ 85 in.3
339 in.3
286 in.3
382 in.3
5. The manufacturer of a concentrated floor
cleaning solution recommends that it be diluted
so that the final mixture is 1 part cleaning
solution to 8 parts water. Mia pours 120 cubic
inches of water into the cylindrical bucket shown
below and then adds the correct amount of
cleaning solution.
/--10 in.---1
Which expression represents how many more
cubic inches of liquid the bucket can hold?
@ 1T. 52 12 - (120 +i 120) 1T. 52 12 + 120 + 120
1T. 102 12- (120 +i. 120) 1T. 102 12- 8 120
6. What is the approximate volume of the paper
water cup shown below? Use 3.14 for 1T.
3 in..
T6 in.
1@ 56.5 in.3
113.1 in.3
108 in.3
226.2 in.3
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7. The top of the grain silo shown below is a
hemisphere. What is the approximate volume of
the silo? Use 3.14 for 7T.
1-16ft-j
T30 ft
1
@ 1910 ft3
23.12 ft 3
2111 ft3
7101 ft3
8. A basketball with circumference 78 centimeters
touches all six sides of its cubical shipping box.
Approximately what percent of the space inside
the box is not occupied by the basketball? Use
3.14 for 7T.
>-ccoCL
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@ 15%
48%
25%
75%
9. Naomi has built part of a sandcastle in the shape.
of a cone with the dimensions shown. To the
nearest cubic inch, what is the volume of this
cone? Use 3.14 for 7T. \
@ 1407 in.3
938 in.3
1081 in.3
299 in.3
10. A three-quarter circle with radius 8 inches is
made into a hat by attaching the edges of the I
cutout. What is the best estimate for the height of
the hat if the diameter of the base is 12inches?
@ 5.3 in. 8 in. 8.9 in. lain.
Date _
11. A cylindrical salt shaker has a radius of
20 millimeters and a height of 90 millimeters.
What is the volume of the salt shaker? Round
your answer to the nearest cubic millimeter.
Use 3.14 for n.
@ 113,040 mm3
28,274mm3
62,424mm3
13,823 mm3
12. The dome of a building is a hemisphere with a
diameter of 50 feet. What is the surface area of
the dome, rounded to the nearest square foot?
Use 3.14 for 7T.
@ 1571 ft2
7854 ft2
3925 ft2
32,725 ft2
13. The pyramid below isa representation of a trellis
for Marla's flowering vine. The base is a regular
hexagon. What is the surface area of the pyramid,
including the base? Round your answer to the
nearest hundredth.
@ 23.38 ft2
95.38 ft2
72.00 ft2
167.38 ft2
14. Archie built a ramp using one rectangular piece
of wood for the top and two triangular pieces for
the sides, as shown below. To the nearest tenth of
a square foot, what is the total surface area of the
plywood Archie used to build the ramp?
3
8.25 ft
@ 44.9 ft2 52.7 ft2 .
65.0ft2 cID 77.4 ft2
California Standards Review and Practice'
. Geometry Standards 35
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Name ~----
California Standards
GeometrY 10.0
Date _
Students compute areas of polygons, including
rectangles, scalene triangles, equilateral triangles, rhombi,
.parallelograms, and trapezoids.
Area
Rectangle A = bh
b is the base.
his the height.
Triangle
b
A = lbh2
b is the base.
his the height.
Equilateral Triangle
b
V3S2A=-
4
s is a side.
h
Rhombus 1A = Zdld2
dI and d2
are the
diagonals.
d II 1
T
IParallelogram A =bh
b is the base.
h is the height.
Trapezoid 1
A = zh(bl + b2)
bl
and b2
are the bases.
h is the height.
California Standards Review and Practice
Geometry Standards
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Example 1
Example 2
Example 3
Area of a Rectangle
The area of arectangular field is4675 square feet. The field is 55feet wide. Find the
length of/the field.
Solution
Let bbe the length and hbe the width ofthe field.
A = bh
4675 = b(55)
b = 85
Formula for area of a rectangleSubstitute 4675 for A and 55 for h.
Simplify.
AnswerThe length of the field is 85 feet.
Area of a Triangle
The base ofatriangle isthree times its height. The area of the triangle is 96 square /
meters. Find the base and height of the triangle.
Solution
Let hrepresent the height of the triangle. Then the base is 3h.
~3hA = lbh
2
96 = ~(3h)(h)
96 = lh22
64 = h2
Formula for area of a triangle
Substitute 96 for A and 3h for b.
Simplify.
Multiply ~achside by ~.
8=h Find the positive squareroot of each side.
AnswerThe height of the triangle is 8meters, and the base is 24 meters.
Area of an Equilateral Triangle
An equilateral triangle has a side length of 12centimeters. What is the area of the
triangle?
Solution
'/-;32 V3(12)2A =7= --4- = 36\13 = 62.4
AnswerThe area of the triangle isabout 62.4 square centimeters.
Area of a Rhombus
Rhombus ABCD has an area of98 square meters. Find AC ifBD = 7 meters.
A~
o c
Formula for area of a
rhombus
Substitute 49 for A and 7
for d1
Simplify.
Answer AC equals 28 meters.
California Standards Review and Practice
.Geometry Standards 37
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Name _ Date ~ _
Example 5 Area of a Parallelogram
A cornfield is shaped like a parallelogram. Find the area of the field in acres. There are
4840 square yards in one acre.
Solution
Find the area of the field in square yards.
A = bh = (250)(150) = 37,500 yd2
Change the units to acres.
~ "~ 1acre.J7,500 J'U 4840ft =7.75 acres
250 yd
150 yd
Use uni't
analysis.
Answer The area of the field is about 7.75 acres.
Example 6 Area of a Trapezoid .
The Art Club needs to buy primer paint so that
members can prime one wall of the school before
painting anew mural. A gallon of primer paint covers
300 square feet. How many gallons of primer paint
should the club buy?
Solution
Find the area of the wall.
A = ~h(bl + b2) = ~(28)(32 + 40) = ~(28)(72)
= 1008 ft2
Determine how many gallons of primer paint are needed.1 gal '
1008~' 300,ff = 3.36 gal Use unit analysis.
Answer Round up so there is enough primer paint. The Art Club should buy
4 gallons of primer paint.
1. Dan and Marie built a deck behind their house.
A sketch of the deck's floor is shown below. They
are planning to waterproof the top of the deck
and need to find its area so they know how many
gallons of water sealant to buy. Ifone gallon of
water sealant covers 150 square feet, how many
gallons of water sealant are required for the
deck? Round your answer to the nearest tenth.
16ft
22 ft
@ 2.7 gal
3.7 gal
2.9 gal
4.0gal
Exercises
32 ft
40 ft
California Standards Review and Practice
Geometry Standards
2. In isosceles trapezoid ABCD, AB = 28 inches
and DC = 48 inches. What additional data does
not provide sufficient information to find the area
of the trapezoid?
A B
L[\DEC
@ the perimeter of the trapezoid
the length ofBC
the measure of LAED
the length ofAE
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3. Chase has a clock in his room with aclock face
in the shape of an equilateral triangle. The area of
the clock face is 16'13 square inches. What is the
length o f a side"of the clock face?
4 in. @ 6 in. 8in. lOin.
4. Brett isusing wallpaper to decorate two walls of
his bedroom. Each wall is 14 feet by 18 feet. If
one roll covers about 27 square feet, how many
rolls will Brett need tocover both walls?
9 rolls
18rolls
@ 10 rolls
19 rolls
5. What is the area of the triangle below?
I 17 I
~
35units2
70 units2
@ 42.5 units2
85units2
6.. The figure below is a square with four congruent
rhombi inside.
11em
Sem
What is the area of the shaded portion?
25 cm2 @ 48 cm2
, 73 cm2 109 cm2
7. What is the area of the quadrilateral below?
150 units2
250 units2
25
@ 200units2
300 units2
Date _
8. What is the area of the parallelogram shown.
below?
It , (7,3)
o
10 units2
20 units2
@ 10.5 units2
21 units2
9. The height of a triangle is 1.5 times the length of
its base. The area of the 'triangle is 75 square feet.
What is the height of the triangle?
10ft
5Y2ft
@ 15 ft
1OY2ft
10. The quadrilateral shown below is a rhombus.
What is the area of L QRS?
Q_I
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Example 1
California Standard
Geometry 11.0
Students determine how changes in dimensions affect the perimeter,
area, and volume of common geometric figures and solids.
Changing Dimensions
Perimeters of Similar Polygons
If two polygons are similar, then the ratio oftheir perimeters is equal to the ratios of their
corresponding side lengths.
Areas of Similar Polygons
If two polygons are similar with the lengths of corresponding sides in the ratio of a: b, then the
ratio of their areas is a2: b2.
Surface Areas of Similar Solids
If two similar solids have a scale factor of a: b,then corresponding areas have a ratio ofa2: b2
Volumes of Similar Solids
If two similar solids have a scale factor of a: b, then corresponding volumes have a ratio of
a3 :b3.
Change Perimeter
A school plans to install a synthetic-turf field to replace a grass one. The rectangular
grass field is 95 yards long and 57 yards wide. The synthetic-turf field will be similar
in shape, but it will be 60 yards wide.
a. Find the scale factor of the old field to the new field.
b. Find the petimeter of the new field.
Solution
a. The scale factor of the old field to the new field is the ratio of the widths, ~6= ~~.b. The perimeter of the original field is 2(57) + 2(95) = 304 yards. Use the perimeter
of similar polygons theorem to find the perimeter x of't.he new field.
304 19
X-20
x = 320
Write a proportion.
Cross multiply and simplify.
Answer The perimeter of the new field is 320 yards.
California Standards Review and Practice
Geometry Standards .
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Example 3
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Change Area
Date _
A large rectangular tabletop is 64 inches long by I..J L
36 inches wide. A smaller tabletop is similar to the large
tabletop. The area of the smaller tabletop is 1296 square I..J L
inches. ~ind the width of the smaller tabletop. 64 in.
Solution
If the area ratio is a2 :b2, then the length ratio is a :b. h I h I
A = 1296 in.2
Write the ratio of known areas. Then simplify.
Find the square root of the area ratio.
Area of smaller tabletop
Area of larger tabletop
Length of smaller tabletop
Length of larger tabletop
1296
2304
V9
ill
9
16
3
4
36 in.
Any length in the smaller tabletop is~, or 0.75, of the corresponding le'ngth in the
larger tabletop. So, the width ofthe smaller tabletop is0.75(36 inches) = 27 inches.
Answer The width of the smaller tabletop is 27 inches.
Change Surface Area
The coffee filters shown are similar with a scale factor of77 : 100. Find the surface
area of the larger coffee filter.
Solution
Write a proportion.------- .......
Surface area ofI a2
Surface area of II= P
47.12 772
Surface area of II = 1002
Surface area of II = 79.47 5=47.12 in.2
- - . . . --------- ...
Answer The surface area of the larger coffee filter is about 79.47 square inches.
Change Volume
The prisms shown are similar with a scale factor of2 : 3.
Find the volume of the larger prism.
Solution
If the two similar solids have ascale
factor of a :b, then the corresponding
volumes have a ratio of a3 : b3.
Volume of smaller prism a3
Volume of larger prism b3
16 23
Volume oflarger prism = 33
V= 16 in.3
Volume of larger prism = 54
Answer The volume ofthe larger prism is 54 cubic inches.
L
,,
California Standards Review and Practice
Geometry Standards 41
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Exercises
1. What is the affect on the area of a circle when the 5. A square-shaped office has a side length
radius is tripled? of 90 feet. The owners want to double the
@ The area is ~ the original area. dimensions of the space. If the area of the
.J existing space is 8100 square feet, what will be
The area is 3 times the original area. the area of the new office space?
The area is unchanged. @ 48,600 ft2
@ The area is 9 times the original area. 32,400 ft2
2. A square has a side length of 5 meters. What is 24,300 ft2
the affect on the perimeter of the square when the @ 16,200 ft2
side length is ~ripled?
@ The perimeter is 1.5 times the original 6. The dimensions of a sphere are increased by a
scale factor of 4. The surface area of the originalperimeter.
sphere is about 314 em 2.What is the surface atea
The perimeter is 3 times the original of the larger sphere?
perimeter.@ 1256 cm2
The perimeter is 6times the original
2512 cm2perimeter.
@ The perimeter is 9 times the original 3768 cm2
perimeter. @ 5024cm2
3. Jessica cans tomatoes in two sizes of jars. The 7. Mr. Gonzalez needs to increase the space he rents
smaller jar has half the dimensions of the larger at a boat yard. He currently rents a rectangular
jar. If the larger jar has a volume of 430 cubic storage space of 6000 cubic feet. If he increases
inches, what is the volume of the smaller jar? the dimensions of the storage space 1.5 times,
@ 53 ~ in.3what will be the volume of the new storage space?
>-
@ 9000 ft3ccoCl.
107 ~in.3E
13,500 ft3a
U
.'= 215 in.3 20,250 ft3
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8. Two spheres are similar with a scale factor of:c
4. Refer to the rectangle below. What is the area of '01 : 3. The volume of the smaller sphere is cthe rectangle after all side lengths are doubled? a.",34 cubic inches. What is the volume of the larger :~
j6cm
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@ 68 in.3illt::
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102 in.3 en:::>@ 96cm2
a
306 in.30u
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192 cm2 >-
@ 918 in.3 ..0
(9
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..cen
@ 576cm2 9. The dimensions of a cone are doubled. Ifthe .~
approximate volume of the cone is 150 cubicCl.
a
U
meters, what is the volume of the larger cone?
@ 300m3 600m3
900m3 .@ 1200m3
California Standards Review and Practice
Geometry Standards
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Example 1
Example 2
Example 3
Date _
Find Interior Angle Measures in a Triangle
Find the measures of the angles of the triangle shown.
Solution
Use the Triangle Sum Theorem to set up and solve an equation. B
80 + 3x + 2x + 15 = 180
5x = 85
x = 17
Substitute the value ofx into the angle expressions.
mLB = 3x = 3(17) = 51
mL C = 2x + 15 = 2(17) + 15 = 49A c
Find Exterior Angle Measures in a Triangle
FindmLQRS.
Solution
Use the Exterior Angle Theorem to set up and solve
an equation.
3x + 50 = 5x + 2
48 = 2x
24 = x
Q
s
(5x+ 2)
Substitute the value ofx into the expression for the exterior angle.
5x + 2 = 5(24) + 2 = 122, so mL QRS = 180 - 122 = 58
Find Side Length
Find the values ofx andy in the diagram.
Solution
From the diagram we know that 3y - 1 = 3x2 + 2.
Using the Converse of the Base Angles Theorem
we know that 3x2 + 2 = y + 9.
So 3x - 1 = y + 9by the Transitive Property of Equality.
Solve fory. Use the value ofy to findx.
3y - 1 =y + 9 3x2 + 2 =y + 9
2y = 10 3x2 + 2 = 5 + 9
y = 5 3x2 = 12
x2 = 4
3V~
y+9
x=2 Find the positive square root.
California Standards Review and Practice
Geometry Standards
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Example 4
Example 5
Example 6'
Classify Triangles
Date _
Can segments with lengths of 9 meters, 10 meters, and 15 meters form a triangle?
If so, would the triangle be acute, right, or obtuse?
Solution
STEP 1 Use theTriangle InequalityTheorem to checkthat the segments can
make a triangle.
9 + 10 = 19
19> 15
9 + 15'= 24
24> 10
10 + 15 = 25
25> 9
STEP 2 Classifythe triangle by comparing the square of the length of the
longest sidewith the sum of squares of the lengths of the shorter
sides.
c2~a2+b2
152 ~ 92 + 102
225 ~ 81 + 100
225> 181
Compare c2 with a2 + b2.
Substitute.
Simplify.
c2.is greater than a2 + b2.
Answer The side lengths 9 meters, 10 meters, and 15 meters form an obtuse triangle.
Find Interior Angle Measures in a Polygon
Find the value ofx i~the diagram.
Solution
The polygon is a pentagon. Use the Polygon Interior Angles
Theorem with n = 5 to write an equation involving x.
Then solve the equation.
mL 1 + mL 2 + mL 3 + mL 4 + mL 5 = (5 - 2) 180
XO + 107
+
90 +
145+
76 =
540
x + 418 = 540
x = 122
Answer;r'hevalue ofx is 122.
Find Exterior Angle Measures in a Polygon
Find the value ofx in the diagram.
Solution
Use the Polygon Exterior Angles Theorem to write
an equation involving x. Then solve the equation.
mL1 + mL2 + mL3 + mL4 = 360
x + 280 = 360
x = 80
Answer The value ofx is 80.
California Standards Review and Practice
Geometry Standards 45.
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Exercises
6
1. An exterior angle of a regular polygon measures
90. What type of figure is the polygon?
@ triangle
square
pentagon
hexagon
2. In the figure below, an exterior angle of the
triangle measures 125.
B
A
Which of the following could not be the measures
of interior angles A and B?
@ 50 and 75
40 and 85
65 and 60
45 and 70
3. The sum of the interior angles of a polygon is
twice the sum of its exterior angles. What type of
figure is the polygon?
@ quadrilateral
hexagon
octagon
nonagon
4. What is the value ofx in the figure below?
@3
4
6
8
California Standards Review and Practice
Geometry Standards
5. In the figure below, BC IIAD.
B C
A
What ismLBCD?
@ 55
120
60
165
6. What type of triangle has side lengths 5 feet,
13 feet, and 16 feet?
@ acute
right
isosceles
obtuse
7. The measures of the interior angles ofa
quadrilateral are xO,3xo,5xo,and 6xo.What is the
measure of the largest interior angle?
@ 24
72
144;>.
180e'
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8. The side lengths ofatriangle are 5, x, and 13. e
What are the values ofx that make the triangle an ~
~
acute triangle? e0E
@ x-..D
@ right triangle@~..cen
equilateral triangle.~
Cl.
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isosceles triangle
scalene triangle,
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Example 1
California Standards
Geometry 13.0
Students prove relationships between angles in polygons by
using properties of complementary, supplementary, vertical,
and exterior angles.
Angles and Polygons
Two angles are complementary angles ifthe sum of their measures is 90.
Each angle is the complement of the other.
Two angles are supplementary angles ifthe sum of their measures is 180.
Each angle is the supplement of the other.
Two adjacent angles are a linear pair if their noncommon
sides are opposite rays. L 1 and L 2 are a linear pair.
Two angles arevertical angles if their sides form two
pairs of opposite rays. In the figure,
L 1and L 3 are vertical angles.
L 2 and L 4 are vertical angles.
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
Vertical Angles Congruence Theorem
Vertical angles are congruent.
Find Angle Measures
Find the values ofx and y in the diagram.
Solution
By the Exterior Angle Theorem:
mLAEB = mLDAE + mLADE
A
x = 144
oBy the Exterior Angle Theorem:mLAEB = mLBCE + mLEBC
1440 = yO + 48
y = 96
Answer The value ofx is 144 and the value ofyis 96.
B
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Geometry Standards 47
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Example 2
Example 3
Find an Angle Measure
Find the value ofx in the diagram.
Solution
By the Base Angles Theorem:
mL TRS = mL TSR = 50
By the Triangle S~m Theorem:
mLSTR + mLTSR + mLTRS = 180
mLSTR + 50 + 50 = 180
mLSTR = 80
By the Vertical Angles. Congruence Theorem:
mL PTQ = mL STR = 80
By the Triangle Sum Theorem:
mLPTQ + mLPQT + mLQPT = 180
80 + 42+ mL QPT = 180
mLQPT=' 58
By the Linear Pair Postulate:
XO + mLQPT = 180
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Exercises
Date _
"1. What is th\!measure ofan exterior angle of a
regular pentagon?
36
45
72
90
2. The figure below shows atower that was built to
support high-voltage power lines. It was designed
as an isosceles triangle. Ifthe side of the tower
meets the ground at a98 angle, what is the
measure of the angle at the top of the tower?
98
8
16
36
41
3. What is the value ofx in the figure below?
28 38 48 58
4. For the figure below, which expression gives the
correct value ofx in terms ofy?
x=3y
x=~3
x = 180 ---':y3
x=y+903
5. What is the value ofx in the figure below?
25
30
45
50
6. What is the value ofx in the figure below?
9
27
54
. 81
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Geometry Standards 49
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