Chapter 5 Resource Masters©Glencoe/McGraw-Hill 248 Glencoe Geometry ALGEBRA In ABC, BF is the angle...

© Glencoe/McGraw-Hill 248 Glencoe Geometry

ALGEBRA In �ABC, B�F� is the angle bisector of �ABC, A�E�, B�F�,and C�D� are medians, and P is the centroid.

1. Find x if DP � 4x � 3 and CP � 30.

2. Find y if AP � y and EP � 18.

3. Find z if FP � 5z � 10 and BP � 42.

4. If m�ABC � x and m�BAC � m�BCA � 2x � 10, is B�F� an altitude? Explain.

ALGEBRA In �PRS, P�T� is an altitude and P�X� is a median.

5. Find RS if RX � x � 7 and SX � 3x � 11.

6. Find RT if RT � x � 6 and m�PTR � 8x � 6.

ALGEBRA In �DEF, G�I� is a perpendicular bisector.

7. Find x if EH � 16 and FH � 6x � 5.

8. Find y if EG � 3.2y � 1 and FG � 2y � 5.

9. Find z if m�EGH � 12z.

COORDINATE GEOMETRY The vertices of �STU are S(0, 1), T(4, 7), and U(8, �3).Find the coordinates of the points of concurrency of �STU.

10. orthocenter 11. centroid 12. circumcenter

13. MOBILES Nabuko wants to construct a mobile out of flat triangles so that the surfacesof the triangles hang parallel to the floor when the mobile is suspended. How canNabuko be certain that she hangs the triangles to achieve this effect?

DI

HF

G

E

S R

P

TX

A

C

F

E

DP

B

Practice Bisectors, Medians, and Altitudes

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

Skills PracticeInequalities and Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2


Less

on

5-2

Determine which angle has the greatest measure.

1. �1, �3, �4 2. �4, �5, �7

3. �2, �3, �6 4. �5, �6, �8

Use the Exterior Angle Inequality Theorem to list all angles that satisfy the stated condition.

5. all angles whose measures are less than m�1

6. all angles whose measures are less than m�9

7. all angles whose measures are greater than m�5

8. all angles whose measures are greater than m�8

Determine the relationship between the measures of the given angles.

9. m�ABD, m�BAD 10. m�ADB, m�BAD

11. m�BCD, m�CDB 12. m�CBD, m�CDB

Determine the relationship between the lengths of the given sides.

13. L�M�, L�P� 14. M�P�, M�N�

15. M�N�, N�P� 16. M�P�, L�P�

83� 57�79�

44�59�

38�LN

P

M

2334

4139

35A

B C

D

1

2 4

6

7

8 93 5

1 2 4 6 7 8

35


Construction ProblemThe diagram below shows segment AB adjacent to a closed region. Theproblem requires that you construct another segment XY to the right of theclosed region such that points A, B, X, and Y are collinear. You are not allowedto touch or cross the closed region with your compass or straightedge.

Follow these instructions to construct a segment XY so that it iscollinear with segment AB.

1. Construct the perpendicular bisector of A�B�. Label the midpoint as point C,and the line as m.

2. Mark two points P and Q on line m that lie well above the closed region.Construct the perpendicular bisector n of P�Q�. Label the intersection oflines m and n as point D.

3. Mark points R and S on line n that lie well to the right of the closedregion. Construct the perpendicular bisector k of R�S�. Label theintersection of lines n and k as point E.

4. Mark point X on line k so that X is below line n and so that E�X� iscongruent to D�C�.

5. Mark points T and V on line k and on opposite sides of X, so that X�T� andX�V� are congruent. Construct the perpendicular bisector � of T�V�. Call thepoint where the line � hits the boundary of the closed region point Y. X�Y�corresponds to the new road.

Q

P

m

k

nD

R E

T

X

V

BAC

S

ExistingRoad

Closed Region(Lake)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2

Skills PracticeIndirect Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3


Less

on

5-3

Write the assumption you would make to start an indirect proof of each statement.

1. m�ABC � m�CBA

2. �DEF � �RST

3. Line a is perpendicular to line b.

4. �5 is supplementary to �6.

PROOF Write an indirect proof.

5. Given: x2 � 8 � 12Prove: x � 2

6. Given: �D � �F.Prove: DE � EF

D F

E

Study Guide and InterventionThe Triangle Inequality

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4


Less

on

5-4

The Triangle Inequality If you take three straws of lengths 8 inches, 5 inches, and 1 inch and try to make a triangle with them, you will find that it is not possible. Thisillustrates the Triangle Inequality Theorem.

Triangle Inequality The sum of the lengths of any two sides of aTheorem triangle is greater than the length of the third side.

The measures of two sides of a triangle are 5 and 8. Find a rangefor the length of the third side.By the Triangle Inequality, all three of the following inequalities must be true.

5 � x � 8 8 � x � 5 5 � 8 � xx � 3 x � �3 13 � x

Therefore x must be between 3 and 13.

Determine whether the given measures can be the lengths of the sides of atriangle. Write yes or no.

1. 3, 4, 6 2. 6, 9, 15

3. 8, 8, 8 4. 2, 4, 5

5. 4, 8, 16 6. 1.5, 2.5, 3

Find the range for the measure of the third side given the measures of two sides.

7. 1 and 6 8. 12 and 18

9. 1.5 and 5.5 10. 82 and 8

11. Suppose you have three different positive numbers arranged in order from least togreatest. What single comparison will let you see if the numbers can be the lengths ofthe sides of a triangle?

BC

A

a

cb

ExercisesExercises

ExampleExample


Distance Between a Point and a Line

Study Guide and Intervention (continued)

The Triangle Inequality

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4

The perpendicular segment from a point toa line is the shortest segment from thepoint to the line.

P�C� is the shortest segment from P to AB��.

The perpendicular segment from a point toa plane is the shortest segment from thepoint to the plane.

Q�T� is the shortest segment from Q to plane N .

Q

TN

B

P

CA

Given: Point P is equidistant from the sides of an angle.

Prove: B�A� � C�A�Proof:1. Draw B�P� and C�P� ⊥ to 1. Dist. is measured

the sides of �RAS. along a ⊥.2. �PBA and �PCA are right angles. 2. Def. of ⊥ lines3. �ABP and �ACP are right triangles. 3. Def. of rt. �

4. �PBA � �PCA 4. Rt. angles are �.5. P is equidistant from the sides of �RAS. 5. Given6. B�P� � C�P� 6. Def. of equidistant7. A�P� � A�P� 7. Reflexive Property8. �ABP � �ACP 8. HL9. B�A� � C�A� 9. CPCTC

Complete the proof.Given: �ABC � �RST; �D � �UProve: A�D� � R�U�Proof:

1. �ABC � �RST; �D � �U 1.

2. A�C� � R�T� 2.

3. �ACB � �RTS 3.

4. �ACB and �ACD are a linear pair; 4. Def. of �RTS and �RTU are a linear pair.

5. �ACB and �ACD are supplementary; 5.�RTS and �RTU are supplementary.

6. 6. Angles suppl. to � angles are �.

7. �ADC � �RUT 7.

8. 8. CPCTC

A

DC

B

R

UT

S

AS C

PB

R

ExampleExample

ExercisesExercises

Skills PracticeInequalities Involving Two Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5


Less

on

5-5

Write an inequality relating the given pair of angles or segment measures.

1. m�BXA, m�DXA

2. BC, DC

Write an inequality relating the given pair of angles or segment measures.

3. m�STR, m�TRU 4. PQ, RQ

5. In the figure, B�A�, B�D�, B�C�, and B�E� are congruent and AC � DE.How does m�1 compare with m�3? Explain your thinking.

6. Write a two-column proof.Given: B�A� � D�A�

BC � DCProve: m�1 � m�2

12

B

A

D

C

12

3

B

AD C

E

95�7 7

85�P RS

Q31

30

22 22

R S

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6

98

3

3

B

A C

D

X

© Glencoe/McGraw-Hill A3 Glencoe Geometry

An

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and

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5-1

5-1

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ALG

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AF

or E

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1–4

,use

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in

form

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o fi

nd

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of

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Fin

d x

and

RT

if S�

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a m

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9x

�18

;R

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120

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ind

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Fif

B�D�

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e bi

sect

or.

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ind

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d IJ

if H�

K�is

an

alt

itu

de o

f �

HIJ

.

x�

3.5;

EF

�13

x�

29;

IJ�

57

ALG

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AF

or E

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5–7

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rmat

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.In

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,P,Q

,an

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are

the

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M�,M�

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x.4

6.F

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d z.

0.7

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and

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and

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and

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2.2

4.If

m�

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AC

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sin

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F�is

an

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32

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6

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EH

�16

an

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H�

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5.

3.5

8.F

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2y�

1 an

d F

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2y�

5.

5

9.F

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m�

EG

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7.5

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of

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TU

are

S(0

,1),

T(4

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and

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.F

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5-1

5-1

Answers (Lesson 5-1)


Skil

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Ineq

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5-2

5-2

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Det

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wh

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th

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1.�

1,�

3,�

42.

�4,

�5,

�7

�1

�4

3.�

2,�

3,�

64.

�5,

�6,

�8

�6

�8

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th

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An

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Ineq

ual

ity

Th

eore

m t

o li

st a

ll

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es t

hat

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.

5.al

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wh

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mea

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ss t

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m�

1

�2,

�3,

�4,

�5,

�7,

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ss t

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m�

9

�2,

�4,

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wh

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s ar

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�1,

�3

8.al

l an

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wh

ose

mea

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s ar

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�8

�1,

�3,

�5

Det

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ine

the

rela

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ship

bet

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n t

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mea

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s of

th

e gi

ven

an

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.

9.m

�A

BD

,m�

BA

D10

.m�

AD

B,m

�B

AD

m�

AB

D�

m�

BA

Dm

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DB

�m

�B

AD

11.m

�B

CD

,m�

CD

B12

.m�

CB

D,m

�C

DB

m�

BC

D�

m�

CD

Bm

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BD

�m

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DB

Det

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the

rela

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ship

bet

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n t

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len

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s of

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e gi

ven

sid

es.

13.L �

M�,L�

P�14

.M�P�

,M�N�

LM

�L

PM

P�

MN

15.M�

N�,N�

P�16

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MN

�N

PM

P�

LP

83�

57�

79�

44�

59�

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BC

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1.�

1,�

3,�

42.

�4,

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�9

�1

�4

3.�

2,�

3,�

74.

�7,

�8,

�10

�7

�10

Use

th

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An

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Ineq

ual

ity

Th

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m t

o li

st

all

angl

es t

hat

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th

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ated

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dit

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.

5.al

l an

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wh

ose

mea

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s ar

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ss t

han

m�

1

�3,

�4,

�5,

�7,

�8

6.al

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wh

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mea

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s ar

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ss t

han

m�

3

�5,

�7,

�8

7.al

l an

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wh

ose

mea

sure

s ar

e gr

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an m

�7

�1,

�3,

�5,

�9

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l an

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wh

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mea

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s ar

e gr

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an m

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�6,

�9

Det

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ine

the

rela

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ship

bet

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n t

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mea

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s of

th

e gi

ven

an

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.

9.m

�Q

RW

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RW

Q10

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RT

W,m

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m�

QR

W�

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WQ

m�

RT

W�

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11.m

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S12

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WQ

R,m

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m�

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m�

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Det

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ine

the

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13.D �

H�,G�

H�14

.D�E�

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DH

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15.E�

G�,F�

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EG

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GD

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EG

17.S

POR

TST

he

figu

re s

how

s th

e po

siti

on o

f th

ree

tree

s on

on

e pa

rt o

f a

Fri

sbee

™ c

ours

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t w

hic

h t

ree

posi

tion

is

the

angl

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twee

n t

he

tree

s th

e gr

eate

st?

2

53 ft

40 ft

3

2

1

37.5

ft

120�

32�

48�

113�

17�

H

DE

F

G

3447

45

44

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S

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12

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9

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12

46

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910

3

5

Pra

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Ineq

ual

itie

s an

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rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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E__

____

____

__P

ER

IOD

____

_

5-2

5-2



An

swer

s

Skil

ls P

ract

ice

Ind

irec

t P

roo

f

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3

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w-H

ill25

9G

lenc

oe G

eom

etry

Lesson 5-3

Wri

te t

he

assu

mp

tion

you

wou

ld m

ake

to s

tart

an

in

dir

ect

pro

of o

f ea

ch s

tate

men

t.

1.m

�A

BC

�m

�C

BA

m�

AB

C

m�

CB

A

2.�

DE

F�

�R

ST

�D

EF

��

RS

T

3.L

ine

ais

per

pen

dicu

lar

to l

ine

b.L

ine

ais

no

t p

erp

end

icu

lar

to li

ne

b.

4.�

5 is

su

pple

men

tary

to

�6.

�5

is n

ot

sup

ple

men

tary

to

�6.

PRO

OF

Wri

te a

n i

nd

irec

t p

roof

.

5.G

iven

:x2

�8

�12

Pro

ve:x

�2

Pro

of:

Ste

p 1

:A

ssu

me

x�

2.S

tep

2:

If x

�2,

then

x2

�4.

Bu

t if

x2

�4,

it f

ollo

ws

that

x2

�8

�12

.T

his

co

ntr

adic

ts t

he

giv

en f

act

that

x2

�8

�12

.S

tep

3:

Sin

ce t

he

assu

mp

tio

n o

f x

�2

lead

s to

a c

on

trad

icti

on

,it

mu

st

be

fals

e.T

her

efo

re, x

�2

mu

st b

e tr

ue.

6.G

iven

:�D

��

F.

Pro

ve:D

E�

EF

Pro

of:

Ste

p 1

:A

ssu

me

DE

�E

F.

Ste

p 2

:If

DE

�E

F,th

en D�

E��

E�F�by

th

e d

efin

itio

n o

f co

ng

ruen

t se

gm

ents

.B

ut

if D�

E��

E�F�,

then

�D

��

Fby

th

e Is

osc

eles

Tri

ang

le T

heo

rem

.T

his

co

ntr

adic

ts t

he

giv

en in

form

atio

n t

hat

�D

��

F.

Ste

p 3

:S

ince

th

e as

sum

pti

on

th

at D

E�

EF

lead

s to

a c

on

trad

icti

on

,it

mu

st b

e fa

lse.

Th

eref

ore

,it

mu

st b

e tr

ue

that

DE

E

F.

DF

E

©G

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ill26

0G

lenc

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eom

etry

Wri

te t

he

assu

mp

tion

you

wou

ld m

ake

to s

tart

an

in

dir

ect

pro

of o

f ea

ch s

tate

men

t.

1.B�

D�bi

sect

s �

AB

C.

B�D�

do

es n

ot

bis

ect

�A

BC

.

2.R

T�

TS

RT

T

S

PRO

OF

Wri

te a

n i

nd

irec

t p

roof

.

3.G

iven

:�4x

�2

��

10P

rove

:x�

3P

roo

f:S

tep

1:

Ass

um

e x

�3.

Ste

p 2

:If

x�

3,th

en �

4x

�12

.Bu

t �

4x

�12

imp

lies

that

�

4x�

2

�10

,wh

ich

co

ntr

adic

ts t

he

giv

en in

equ

alit

y.S

tep

3:

Sin

ce t

he

assu

mp

tio

n t

hat

x�

3 le

ads

to a

co

ntr

adic

tio

n,

it m

ust

be

tru

e th

at x

�3.

4.G

iven

:m�

2 �

m�

3 �

180

Pro

ve: a

⁄|| bP

roo

f:S

tep

1:

Ass

um

e a

|| b.

Ste

p 2

:If

a|| b

,th

en t

he

con

secu

tive

inte

rio

r an

gle

s �

2 an

d �

3 ar

esu

pp

lem

enta

ry.T

hu

s m

�2

�m

�3

�18

0.T

his

co

ntr

adic

ts t

he

giv

en s

tate

men

t th

at m

�2

�m

�3

18

0.S

tep

3:

Sin

ce t

he

assu

mp

tio

n le

ads

to a

co

ntr

adic

tio

n,t

he

stat

emen

t a

|| bm

ust

be

fals

e.T

her

efo

re,a

⁄|| bm

ust

be

tru

e.

5.PH

YSI

CS

Sou

nd

trav

els

thro

ugh

air

at

abou

t 34

4 m

eter

s pe

r se

con

d w

hen

th

ete

mpe

ratu

re i

s 20

°C.I

f E

nri

que

live

s 2

kilo

met

ers

from

th

e fi

re s

tati

on a

nd

it t

akes

5

seco

nds

for

th

e so

un

d of

th

e fi

re s

tati

on s

iren

to

reac

h h

im,h

ow c

an y

ou p

rove

indi

rect

ly t

hat

it

is n

ot 2

0°C

wh

en E

nri

que

hea

rs t

he

sire

n?

Ass

um

e th

at it

is 2

0°C

wh

en E

nri

qu

e h

ears

th

e si

ren

,th

en s

ho

w t

hat

at

this

tem

per

atu

re it

will

tak

e m

ore

th

an 5

sec

on

ds

for

the

sou

nd

of

the

sire

n t

o r

each

him

.Sin

ce t

he

assu

mp

tio

n is

fal

se,y

ou

will

hav

e p

rove

dth

at it

is n

ot

20°C

wh

en E

nri

qu

e h

ears

th

e si

ren

.

1 23

a b

Pra

ctic

e (

Ave

rag

e)

Ind

irec

t P

roo

f

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Th

e Tr

ian

gle

Ineq

ual

ity

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4

5-4

©G

lenc

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w-H

ill26

3G

lenc

oe G

eom

etry

Lesson 5-4

The

Tria

ng

le In

equ

alit

yIf

you

tak

e th

ree

stra

ws

of l

engt

hs

8 in

ches

,5 i

nch

es,a

nd

1 in

ch a

nd

try

to m

ake

a tr

ian

gle

wit

h t

hem

,you

wil

l fi

nd

that

it

is n

ot p

ossi

ble.

Th

isil

lust

rate

s th

e T

rian

gle

Ineq

ual

ity

Th

eore

m.

Tria

ng

le In

equ

alit

yT

he s

um o

f th

e le

ngth

s of

any

tw

o si

des

of a

Th

eore

mtr

iang

le is

gre

ater

tha

n th

e le

ngth

of

the

third

sid

e.

Th

e m

easu

res

of t

wo

sid

es o

f a

tria

ngl

e ar

e 5

and

8.F

ind

a r

ange

for

the

len

gth

of

the

thir

d s

ide.

By

the

Tri

angl

e In

equ

alit

y,al

l th

ree

of t

he

foll

owin

g in

equ

alit

ies

mu

st b

e tr

ue.

5 �

x�

88

�x

�5

5 �

8 �

xx

�3

x�

�3

13 �

x

Th

eref

ore

xm

ust

be

betw

een

3 a

nd

13.

Det

erm

ine

wh

eth

er t

he

give

n m

easu

res

can

be

the

len

gth

s of

th

e si

des

of

atr

ian

gle.

Wri

te y

esor

no.

1.3,

4,6

yes

2.6,

9,15

no

3.8,

8,8

yes

4.2,

4,5

yes

5.4,

8,16

no

6.1.

5,2.

5,3

yes

Fin

d t

he

ran

ge f

or t

he

mea

sure

of

the

thir

d s

ide

give

n t

he

mea

sure

s of

tw

o si

des

.

7.1

and

6 8.

12 a

nd

18

5 �

n�

76

�n

�30

9.1.

5 an

d 5.

5 10

.82

and

8

4 �

n�

774

�n

�90

11.S

upp

ose

you

hav

e th

ree

diff

eren

t po

siti

ve n

um

bers

arr

ange

d in

ord

er f

rom

lea

st t

ogr

eate

st.W

hat

sin

gle

com

pari

son

wil

l le

t yo

u s

ee i

f th

e n

um

bers

can

be

the

len

gth

s of

the

side

s of

a t

rian

gle?

Fin

d t

he

sum

of

the

two

sm

alle

r n

um

ber

s.If

th

at s

um

is g

reat

er t

han

th

ela

rges

t n

um

ber

,th

en t

he

thre

e n

um

ber

s ca

n b

e th

e le

ng

ths

of

the

sid

eso

f a

tria

ng

le.

BC

A

a

cb

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

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ill26

4G

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eom

etry

Dis

tan

ce B

etw

een

a P

oin

t an

d a

Lin

e

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Th

e Tr

ian

gle

Ineq

ual

ity

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4

5-4

Th

e pe

rpen

dicu

lar

segm

ent

from

a p

oin

t to

a li

ne

is t

he

shor

test

seg

men

t fr

om t

he

poin

t to

th

e li

ne.

P�C�

is t

he s

hort

est

segm

ent

from

Pto

AB

�� .

Th

e pe

rpen

dicu

lar

segm

ent

from

a p

oin

t to

a pl

ane

is t

he

shor

test

seg

men

t fr

om t

he

poin

t to

th

e pl

ane.

Q�T�

is t

he s

hort

est

segm

ent

from

Qto

pla

ne N

.

Q TN

B

P CA G

iven

:Poi

nt

Pis

eq

uid

ista

nt

from

th

e si

des

of

an

an

gle.

Pro

ve:B �

A��

C�A�

Pro

of:

1.D

raw

B �P�

and

C�P�

⊥to

1.

Dis

t.is

mea

sure

d th

e si

des

of �

RA

S.

alon

g a

⊥.

2.�

PB

Aan

d �

PC

Aar

e ri

ght

angl

es.

2.D

ef.o

f ⊥

lin

es3.

�A

BP

and

�A

CP

are

righ

t tr

ian

gles

.3.

Def

.of

rt.�

4.�

PB

A�

�P

CA

4.R

t.an

gles

are

�.

5.P

is e

quid

ista

nt

from

th

e si

des

of �

RA

S.

5.G

iven

6.B �

P��

C�P�

6.D

ef.o

f eq

uid

ista

nt

7.A �

P��

A�P�

7.R

efle

xive

Pro

pert

y8.

�A

BP

��

AC

P8.

HL

9.B �

A��

C�A�

9.C

PC

TC

Com

ple

te t

he

pro

of.

Giv

en:�

AB

C�

�R

ST

;�D

��

UP

rove

:A �D�

�R�

U�P

roof

:

1.�

AB

C�

�R

ST

;�D

��

U1.

Giv

en2.

A�C�

�R�

T�2.

CP

CT

C3.

�A

CB

��

RT

S3.

CP

CT

C4.

�A

CB

and

�A

CD

are

a li

nea

r pa

ir;

4.D

ef.o

f lin

ear

pai

r�

RT

San

d �

RT

Uar

e a

lin

ear

pair

.

5.�

AC

Ban

d �

AC

Dar

e su

pple

men

tary

;5.

Lin

ear

pai

rs a

re s

up

pl.

�R

TS

and

�R

TU

are

supp

lem

enta

ry.

6.�

AC

D�

�R

TU

6.A

ngl

es s

upp

l.to

�an

gles

are

�.

7.�

AD

C�

�R

UT

7.A

AS

8.A�

D��

R�U�

8.C

PC

TC

A DC

B

R UT

S

AS

CPB

R

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Ineq

ual

itie

s In

volv

ing

Tw

o T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-5

5-5

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1G

lenc

oe G

eom

etry

Lesson 5-5

Wri

te a

n i

neq

ual

ity

rela

tin

g th

e gi

ven

pai

r of

an

gles

or

seg

men

t m

easu

res.

1.m

�B

XA

,m�

DX

A

m�

BX

A�

m�

DX

A

2.B

C,D

C

BC

�D

C

Wri

te a

n i

neq

ual

ity

rela

tin

g th

e gi

ven

pai

r of

an

gles

or

segm

ent

mea

sure

s.

3.m

�S

TR

,m�

TR

U4.

PQ

,RQ

m�

ST

R�

m�

TR

UP

Q�

RQ

5.In

th

e fi

gure

,B�A�

,B�D�

,B�C�

,an

d B�

E�ar

e co

ngr

uen

t an

d A

C�

DE

.H

ow d

oes

m�

1 co

mpa

re w

ith

m�

3? E

xpla

in y

our

thin

kin

g.

m�

1 �

m�

3;F

rom

th

e g

iven

info

rmat

ion

an

d t

he

SS

S In

equ

alit

y T

heo

rem

,it

follo

ws

that

in �

AB

Can

d �

DB

Ew

e h

ave

m�

AB

C�

m�

DB

E.S

ince

m

�A

BC

�m

�1

�m

�2

and

m�

DB

E�

m�

3 �

m�

2,it

fo

llow

s th

at m

�1

�m

�2

�m

�3

�m

�2.

Su

btr

act

m�

2 fr

om

eac

h s

ide

of

the

last

ineq

ual

ity

to g

et

m�

1 �

m�

3.

6.W

rite

a t

wo-

colu

mn

pro

of.

Giv

en:B �

A��

D�A�

BC

�D

CP

rove

:m�

1 �

m�

2

Pro

of:

Sta

tem

ents

Rea

son

s

1.B�

A��

D�A�

1.G

iven

2.B

C�

DC

2.G

iven

3.A�

C��

A�C�

3.R

efle

xive

Pro

per

ty4.

m�

1 �

m�

24.

SS

S In

equ

alit

y

1 2

B

A

D

C

12

3

B

AD

C

E

95�

77

85�

PR

SQ31

30

2222

RS

UT

6

98

3

3

B

AC

D

X

©G

lenc

oe/M

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ill27

2G

lenc

oe G

eom

etry

Wri

te a

n i

neq

ual

ity

rela

tin

g th

e gi

ven

pai

r of

an

gles

or

segm

ent

mea

sure

s.

1.A

B,B

K2.

ST

,SR

AB

�B

KS

T�

SR

3.m

�C

DF

,m�

ED

F4.

m�

R,m

�T

m�

CD

F�

m�

ED

Fm

�R

�m

�T

5.W

rite

a t

wo-

colu

mn

pro

of.

Giv

en:G

is t

he

mid

poin

t of

D �F�

.m

�1

�m

�2

Pro

ve:E

D�

EF

Pro

of:

Sta

tem

ents

Rea

son

s

1.G

is t

he

mid

po

int

of

D�F�.

1.G

iven

2.D�

G��

F�G�2.

Def

init

ion

of

mid

po

int

3.E�

G��

E�G�

3.R

efle

xive

Pro

per

ty4.

m�

1 �

m�

24.

Giv

en5.

ED

�E

F5.

SA

S In

equ

alit

y

6.TO

OLS

Reb

ecca

use

d a

spri

ng

clam

p to

hol

d to

geth

er a

ch

air

leg

she

repa

ired

wit

h w

ood

glu

e.W

hen

sh

e op

ened

th

e cl

amp,

she

not

iced

th

at t

he

angl

e be

twee

n t

he

han

dles

of

the

clam

pde

crea

sed

as t

he

dist

ance

bet

wee

n t

he

han

dles

of

the

clam

pde

crea

sed.

At

the

sam

e ti

me,

the

dist

ance

bet

wee

n t

he

grip

pin

g en

ds o

f th

e cl

amp

incr

ease

d.W

hen

sh

e re

leas

ed t

he

han

dles

,th

e di

stan

ce b

etw

een

th

e gr

ippi

ng

end

of t

he

clam

p de

crea

sed

and

the

dist

ance

bet

wee

n t

he

han

dles

in

crea

sed.

Is t

he

clam

p an

exa

mpl

e of

th

e S

AS

or

SS

S I

neq

ual

ity?

SA

S In

equ

alit

y

12

DF

E

G

2021

RT

S

JK

1414

14

13

12C

F

E

D

( x �

3) �

( x �

3) �

1010

RT

S

Q

40�

30�

60�

AK

M

B

Pra

ctic

e (

Ave

rag

e)

Ineq

ual

itie

s In

volv

ing

Tw

o T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-5

5-5


Chapter 5 Resource Masters©Glencoe/McGraw-Hill 248 Glencoe Geometry ALGEBRA In ABC, BF is the angle...

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Transcript of Chapter 5 Resource Masters©Glencoe/McGraw-Hill 248 Glencoe Geometry ALGEBRA In ABC, BF is the angle...