geometry standards workbook

8/13/2019 Geometry Standards Workbook

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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.


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


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


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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.




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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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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.


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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


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.




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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.


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




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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


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.


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


/:"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


8 Geometry Standards

5. In the figure below, HI bisects LKHI and LKII.

H

K

J


/:"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


/:"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.


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


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.




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Exercises

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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


21/81

Name ~ __


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.


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.


1.

L 3 and L5 are supplementary.

.m

n



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22/81

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Name ,Date

Example

If a quadrilateral is a parallelogram, then its opposite Q, II ,R

sides are congruent.


p

If a quadrilateral is a parallelogram, then its opposite Q, } \I,R

angles are congruent.


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.


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


23/81

Name ~ __

T'ildpe~p.!ds,and' 7,'x .... , >F I

t.+.. it/,i!,

I


24/81

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Name ~ __ Date _

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



25/81

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?



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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26/81

Name _ Date __


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


27/81

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.


Geometry Standards

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28/81

Name _ Date _

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29/81

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


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

--


30/81

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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




31/81

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.


Geometry Standards

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32/81

Name ~------ Date _

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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



33/81

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


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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Name _

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


35/81

Name ~----


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.


Geometry Standards

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36/81

Name __ Date _

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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.


.Geometry Standards 37


37/81

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


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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38/81

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Name ___

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


39/81

Name' ~ _ Date _

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.


Geometry Standards .

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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

,,




41/81

Name ~ _ Date _

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

~~c

@ 860 in.3 @ 27,000ft3 .8

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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 :~

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@ 918 in.3 ..0

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@ 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


Geometry Standards


42/81

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43/81

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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.


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.


Geometry Standards 45.


45/81

Name __ Date _

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


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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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,


46/81

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Name ~ _ Date __ ,..-- _

Example 1


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




47/81

Name __ Date _

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


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


25

30

45

50


9

27

54

. 81




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geometry standards workbook

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