Geometry

4-7 Congruence in CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Overlapping Triangles
MG-ASFRST.9B1.52 .GUs-SeRcoT.n2g.r5u eUnsce c.o. n. gcriuternicaeto. .s.oclrvieteria to
psorolvbelepmrosbalenmd spraonvdeprreolavteiorneslahtiiposnisnhigpesoimn egteroicmfiegturirces.
MfigPur1e,s.MP 3, MP 4
MP 1, MP 3, MP 4

Objectives To identify congruent overlapping triangles
To prove two triangles congruent using other congruent triangles

Do all the triangles An assignment for your graphic design class is to
make you dizzy? make a colorful design using triangles. How many
Try to see each triangles are in your design? Explain how you
one. Then learn count them.
some tricks that
may help you.

MATHEMATICAL

PRACTICES In the Solve It, you located individual triangles among a jumble of triangles. Some
triangle relationships are difficult to see because the triangles overlap.

Essential Understanding  You can sometimes use the congruent corresponding
parts of one pair of congruent triangles to prove another pair of triangles congruent.
This often involves overlapping triangles.

Overlapping triangles may have a common side or angle. You can simplify your work
with overlapping triangles by separating and redrawing the triangles.

Problem 1 Identifying Common Parts

What common angle do △ACD and A E
△ECB share?
How can you see an C
individual triangle in Separate and redraw △ACD and △ECB. BD D
order to redraw it? C
Use your finger to AE
trace along the lines DB
connecting the three
vertices. Then cover up CC
any untraced lines.
hsm11gmse_0407_t05049.ai
The common angle is ∠C.

hsm11gmse_0407_t05050.ai B
Got It? 1. a. What is the common side in △ABD and A

△DCA?
b. What is the common side in △ABD and

△BAC ?

Lesson 4-7  Congruence in Overlapping Triangles 265

hsm11gmse_0407_t05051.ai

Proof Problem 2 Using Common Parts Z Y
Given:  ∠ZXW ≅ ∠YWX, ∠ZWX ≅ ∠YXW W X
Prove:  ZW ≅ YX

• ∠ZXW ≅ ∠YWX and ∠ZWX ≅ ∠YXW
• The diagram shows that △ZWX and △YXW are

overlapping triangles.

A diagram of the triangles separated Y hsm11gmse_0407_t05052
Z
Show △ZWX ≅ △YXW . Then use
corresponding parts of congruent
triangles to prove ZW ≅ YX.

W XW X

∠ZXW ≅ ∠YWX
Given

hWsXm≅11WgXmse_0407_t0505△3ZWX ≅ △YXW ZW ≅ YX
Corresp. parts of
Reflexive Prop. of ≅ ASA
≅ are ≅.
∠ZWX ≅ ∠YXW
B
Given
E
Got It? 2. Given:  △ACD ≅ △BDC A D

Prove:  CE ≅ DE

hsm11gmse_0407_t05054 C

Proof Problem 3 Using Two Pairs of Triangles
Given:  In the origami design, E is the midphosinmt o1f1AgCmasned_D04B0. 7_t05055
How do you choose
another pair of Prove:  △GED ≅ △JEB
triangles to help in
your proof? Proof:  E is the midpoint of AC and DB, so AE ≅ CE A E B
Look for triangles that and DE ≅ BE. ∠AED ≅ ∠CEB because vertical G J
share parts with △GED angles are congruent. Therefore, △AED ≅ △CEB D C
and △JEB and that you by SAS. ∠D ≅ ∠B because corresponding
can prove congruent.
In this case, first prove parts of congruent triangles are congruent.
△AED ≅ △CEB.
∠GED ≅ ∠JEB because vertical angles are
congruent. Therefore, △GED ≅ △JEB by ASA.

266 Chapter 4  Congruent Triangles

Got It? 3. Given:  PS ≅ RS, ∠PSQ ≅ ∠RSQ P Q
Prove:  △QPT ≅ △QRT R

T

S

When several triangles overlap and you need to use one pair of congruent triangles to

prove another pair congruent, you may find it helpful tohsdmra1w1agdmiasgera_m04o0f 7ea_cth05pa0i5r 6of

triangles.

Proof Problem 4 Separating Overlapping Triangles

Which triangles are Given:  CA ≅ CE , BA ≅ DE C
useful here?
If △BXA ≅ △DXE, Prove:  BX ≅ DX
then BX ≅ DX
because they are BXD
corresponding parts. If
△BAE ≅ △DEA, B XD AE D
you will have enough B E
information to show
△BXA ≅ △DXE. AE A EA

Statements Reasons

1) BA ≅ DE 1) Given

2h)s mCA11≅gmCEse_0407_t05057 2) Given

3) ∠CAE ≅ ∠CEA 3) Base ⦞ of an isosceles △ are ≅.

4) AE ≅ AE 4) Reflexive Property of ≅

5) △BAE ≅ △DEA 5) SAS

6) ∠ABE ≅ ∠EDA 6) Corresp. parts of ≅ △s are ≅.

7) ∠BXA ≅ ∠DXE 7) Vertical angles are ≅.

8) △BXA ≅ △DXE 8) AAS

9) BX ≅ DX 9) Corresp. parts of ≅ △s are ≅.

Got It? 4. Given:  ∠CAD ≅ ∠EAD, ∠C ≅ ∠E A BC
Prove:  BD ≅ FD D

FE

hsm11gmse_0407_t05058 267

Lesson 4-7  Congruence in Overlapping Triangles

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

Identify any common angles or sides. 5. Reasoning  In Exercise 1, both triangles have vertices

1. △MKJ and △LJK 2. △DEH and △DFG J and K. Are ∠J and ∠K common angles for △MKJ
M and △LJK ? Explain.
L GE
6. Error Analysis  In the diagram,
D △PSY ≅ △SPL. Based on that P S

JK HF fact, your friend claims that R
△PRL R △SRY . Explain why

Separate and redraw the overlapping triangles. Label your friend is incorrect. L Y

the vertices. 7. In the figure below, which pair of triangles could

3. Nhsm11gmse_040P7_t0540. 5h9sm11gmBse_0407_t05060 you prove congruent first in order to prove that
△ACD ≅ △CAB? Explain.

DE A B hsm11gmse_0407_t05063

M QA C

E

DC

hsm11gmse_0407_t05061hsm11gmse_0407_t05062

Practice and Problem-Solving Exercises MATHEMATICAL

hsPmRA1C1TgICmEsSe_0407_t05065

A Practice In each diagram, the red and blue triangles are congruent. Identify their See Problem 1.
common side or angle.

8. K P 9. D E 10. X T

LN W
M GF

YZ

Separate and redraw the indicated triangles. Identify any common angles

or sides. hsm11gmse_0407_t02768

11. h△sPmQ1S1agnmd s△eQ_P0R4 07_t02767 12. △ACB and △PRB 13. △hsJKmL1a1ngdm△sMe_L0K407_t02769

P Q A P KL

T RC O
SR B JM

hsm11gmse_0407_t02771 hsm11gmse_0407_t02772

hsm11gmse_0407_t02770

268 Chapter 4  Congruent Triangles

14. Developing Proof  Complete the flow proof. P See Problem 2.

Given:  ∠T ≅ ∠R, PQ ≅ PV

Prove:  ∠PQT ≅ ∠PVR VQ

∠T ≅ ∠R S
a. TR

∠TPQ ≅ ∠RPV △TPQ ≅ △RPV ∠PQT ≅ ∠PVR
b.
d. hsm11gmse_040e7. _t02773
PQ ≅ PV
c.

15. Given:  RS ≅ UT , RT ≅ US 16. Given:  QD ≅ UA, ∠QDA ≅ ∠UAD
Proo f Prove:  △RST ≅ △UTS P roo f Prove:  △QDA ≅ △UAD

hsm11S gmse_T0407_t02774 Q U

RM U R

WV DA

17. Given:  ∠1 ≅ ∠2, ∠3 ≅ ∠4 18. Given:  AD ≅ ED, See Problems 3 and 4.
Proo f Prove:  △QET ≅ △QEU
Proof D is the midpoint of BF
hsm11gmseT_0407_t02775
Phrsomve1:1  △gmADseC_≅04△0E7D_Gt02776 A

G
1
Q 3 E2 B
4
F B
E D
U C

B Apply 19. Think About a Plan  In the diagram at the right, ∠V ≅ ∠S, PQ

VhsUm≅11SgTm,saen_d04P0S7_≅t0Q27V7.7W.ahi ich two triangles are congruent
hsm11gmse_0W407_Xt0277R8.ai
by SAS? Explain.

• How can you use a new diagram to help you identify

the triangles? VU TS
• What do you need to prove triangles congruent by SAS?

hsm11gmse_0407_t02779.ai

Lesson 4-7  Congruence in Overlapping Triangles 269

STEM 20. Clothing Design  The figure at the right is part of a clothing G
design pattern, and it has the following relationships.
E 7
• GC # AC 8
• AB # BC B H J9
• AB } DE } FG 4
• m∠A = 50
• △DEC is isosceles with base DC. I 5
12 36
a. Find the measures of all the numbered angles in the figure. A DF
b. Suppose AB ≅ FC. Name two congruent triangles and C

explain how you can prove them congruent.

21. Given:  AC ≅ EC , CB ≅ CD 22. Given:  QT # PR, QT bisects PR,
Proo f Prove:  ∠A ≅ ∠E Proof
C QT bisects ∠VQS

BD Prove:  VQ ≅ SQ 
AF E
P Q R

VS

T

Openh-Esmnd1e1dg  Dmrasew_t0h4e0d7ia_gtr0a2m7d8e0s.cariibed. hsm11gmse_0407_t02781.a

23. Draw a vertical segment on your paper. On the right side of the segment draw two
triangles that share the vertical segment as a common side.

24. Draw two triangles that have a common angle.

25. Given:  TE ≅ RI , TI ≅ RE, 26. Given:  AB # BC , DC # BC,
Proof ∠TDI and ∠ROE are right ⦞ Proof
AC ≅ DB

Prove:  TD ≅ RO Prove:  AE ≅ DE

    T I   A D
O
E

D R BC
E

C Challenge 27. Identify a pair of overlapping congruent triangles in the A B

diagram. Then use the given information to write a proof F

to shhsomw1th1agtmthseetr_i0an4g0l7es_atr0e2c7o8n2gruent. hsm11gmse_0D407_t02E783

Given:  AC ≅ BC, ∠A ≅ ∠B

C

hsm11gmse_0407_t02784

270 Chapter 4  Congruent Triangles

28. Reasoning  Draw a quadrilateral ABCD with AB } DC, AD } BC, and diagonals AC
Proof and DB intersecting at E. Label your diagram to indicate the parallel sides.

a. List all the pairs of congruent segments in your diagram.
b. Writing  Explain how you know that the segments you listed are congruent.

Standardized Test Prep

SAT/ACT 29. According to the diagram at the right, which statement is true? G F
△DEH ≅ △GFH by AAS E
△DEH ≅ △GFH by SAS H
△DEF ≅ △GFE by AAS
△DEF ≅ △GFE by SAS D
143
30. △ABC is isosceles with base AC. If m∠C = 37, what is m∠B?
37 74 106

31. Which word correctly completes the statement “All ? angles are congruent”?
hsm11 cgormressep_o0n4di0n7g_t05015
adjacent supplementary right

RExetsepnodnesde 32. I an. tChoepfyigtuhree,dLiaJg}raGmK. aTnhdenMmisarthkeyomuirddpioaignrtaomf LG. J G
with
the given information. M
K
b. Prove △LJM ≅ △GKM. L

c. Can you prove that △LJM ≅ △GKM another way? Explain.

Mixed Review

33. Developing Proof  Complete the paragraph proof. hsm11gSmeeseL_e0ss4o0n74_-t60.5018

Given:  AB ≅ DB, ∠A and ∠D are right angles D
C
Prove:  △ABC ≅ △DBC
BA
Proof:  You are given that AB ≅ DB and ∠A and ∠D are right
angles. △ABC and △DBC are a. ? triangles by the definition of
b. ? triangle. BC ≅ BC by the c. ? Property of Congruence.
△ABC ≅ △DBC by the d. ? Theorem.

34. Constructions  Draw a line p and a point M not on p. Then construct line n See Lesson 3-6.
through M so that n # p.

G et Ready!  To prepare for Lesson 5-1, do Exercises 35–37. hsm11gmse_0407_t02785

Find the coordinates of the midpoint of AB. See Lesson 1-7.
37. A(7, 10), B( -5, -8)
35. A( -2, 3), B(4, 1) 36. A(0, 5), B(3, 6)

Lesson 4-7  Congruence in Overlapping Triangles 271

4 Pull It All Together

RMANCPERFOE TASKCompleting the Performance Task

To solve these Look back at your results from the Apply What You’ve Learned sections in Lessons 4-1,
problems you 4-3, and 4-4. Use the work you did to complete the following.
will pull together 1. Solve the problem in the Task Description on page 217 by estimating the distance XY
many concepts
and skills that across the gorge. Show all your work and explain each step of your solution.
you have studied
about congruent 2. Reflect  Choose one of the Mathematical Practices below and explain how you
triangles. applied it in your work on the Performance Task.

MP 1: Make sense of problems and persevere in solving them.

MP 3: Construct viable arguments and critique the reasoning of others.

MP 7: Look for and make use of structure.

On Your Own

Caitlin wants to estimate the distance across a pond. She begins at one end of the
pond, shown as point R in the diagram below. She turns away from the pond,
walks 300 ft in a straight line, and marks point S. She walks another 300 ft in
the same direction and marks point T.
Next, Caitlin goes over to point U at the other end of the pond. She then measures
the distance as she walks on a straight line to point S and finds that this distance
is 340 ft. She continues another 340 ft in the same direction and marks point V.

RU

S

VT

a. Copy the diagram and label it with the given information.
b. What additional measurement should Caitlin determine to estimate the distance

RU across the pond? Justify your answer.

272 Chapter 4  Pull It All Together

4 Chapter Review

Connecting and Answering the Essential Questions

1 Visualization Congruent Figures (Lesson 4-1) D
You can identify BE
corresponding parts of
congruent triangles by △ABC ≅ △DEF A
visualizing the figures
placed on top of each CF
other.
Using Corresponding Parts
2 Reasoning
and Proof (PLreosvsionngsT 4r-i2a,n 4g-l3e,shaCnsodmn4g1-r16u)genmt se_04cr_t05067 of Congruent Triangles
You can show two
triangles are congruent Side-Side-Side (SSS), Side-Angle-Side (SAS), (Lessons 4-4 and 4-7)
by proving that certain Angle-Side-Angle (ASA), Angle-Angle-Side (AAS),
relationships exist Hypotenuse-Leg (HL) If△LMN ≅ △QRS LM ≅ QR
between three pairs of M MN ≅ RS
corresponding parts. NL ≅ SQ
L N Q ∠L ≅ ∠Q
3 Reasoning S ∠M ≅ ∠R
and Proof ∠N ≅ ∠S
You can tell whether R
a triangle is isosceles
or equilateral by Isosceles and Equilateral Triangles hsm11gmse_04cr_t05068
looking at the number
of congruent angles (Lesson 4-5)
or sides.
• The base angles of an isosceles triangle
are congruent.

• All equilateral triangles are equiangular.
• A ll equiangular triangles are equilateral.

Chapter Vocabulary • corollary (p. 252) • vertex angle of an isosceles triangle
• hypotenuse (p. 258) (p. 250)
• base angles of an isosceles triangle • legs of an isosceles triangle (p. 250)
(p. 250) • legs of a right triangle (p. 258)

• base of an isosceles triangle (p. 250)
• congruent polygons (p. 219)

Choose the correct term to complete each sentence.
1. The two congruent sides of an isosceles triangle are the ? .
2. The side opposite the right angle of a right triangle is the ? .
3. A ? to a theorem is a statement that follows immediately from the theorem.
4. ? have congruent corresponding parts.

Chapter 4  Chapter Review 273

4-1  Congruent Figures

Quick Review Exercises

Congruent polygons have congruent corresponding RSTUV @ KLMNO. Complete the congruence statements.
parts. When you name congruent polygons, always list
corresponding vertices in the same order. 5. TS ≅ ? 6. ∠N ≅ ?

Example 7. LM ≅ ? 8. VUTSR ≅ ?

HIJK @ PQRS. Write all possible congruence statements. WXYZ ≅ PQRS. Find each measure or length.
The order of the parts in the congruence statement tells you
which parts correspond. W Z RQ
Sides: HI ≅ PQ, IJ ≅ QR, JK ≅ RS, KH ≅ SP 80Њ 100Њ 5
Angles: ∠H ≅ ∠P, ∠I ≅ ∠Q, ∠J ≅ ∠R, ∠K ≅ ∠S
145Њ 8.6 35Њ
10 P
X 3Y S

9. m∠P 10. QR 11. WX

12. m∠Z 13. m∠X 14. m∠R

hsm11gmse_04cr_t05080

4-2 and 4-3  Triangle Congruence by SSS, SAS, ASA, and AAS

Quick Review Exercises

You can prove triangles congruent with limited information 15. In △HFD, what angle is included between DH
about their congruent sides and angles. and DF ?

Postulate or Theorem You need 16. In △OMR, what side is included between
∠M  and ∠R?
Side-Side-Side (SSS) three sides

Side-Angle-Side (SAS) two sides and an Which postulate or theorem, if any, could you use to
included angle prove the two triangles congruent? If there is not enough
information to prove the triangles congruent, write not
Angle-Side-Angle (ASA) two angles and an enough information.
included side

Angle-Angle-Side (AAS) two angles and a 17. 18.
nonincluded side

Example 19. 20.

What postulate would you use to prove the triangles hsm11gmse_04cr_t05082hsm11gmse_04cr_t05083
congruent?

You know that three sides are
congruent. Use SSS.

hsm11gmse_04cr_t050h81sm11gmse_04cr_t05084hsm11gmse_04cr_t05085

274 Chapter 4  Chapter Review

4-4  Using Corresponding Parts of Congruent Triangles

Quick Review Exercises

Once you know that triangles are congruent, you can make How can you use congruent triangles to prove the
conclusions about corresponding sides and angles because, statement true?
by definition, corresponding parts of congruent triangles
are congruent. You can use congruent triangles in the 21. TV ≅ YW 22. BE ≅ DE
proofs of many theorems.
V W C
BD
Example

How can you use congruent triangles to prove jQ @ jD? T YX E
WK
D 23. ∠B ≅ ∠D 24. KN ≅ ML

Q EV C K L

Since △QWE ≅ △DVK by AAS, you know that ∠Q ≅ ∠D hsm11gmse_04cr_t05087hsm11gmse_04cr_t05088
because corresponding parts of congruent triangles
are congruent. B ED N M

hsm11gmse_04cr_t05086

4-5  Isosceles and Equilateral Triangles hsm11gmse_04cr_t05089hsm11gmse_04cr_t05090

Quick Review Exercises

If two sides of a triangle are congruent, then the Algebra  Find the values of x and y.
angles opposite those sides are also congruent by
the Isosceles Triangle Theorem. If two angles of a 25. 26.
triangle are congruent, then the sides opposite the
angle are congruent by the Converse of the Isosceles x 50Њ 4 xЊ yЊ
Triangle Theorem. 125Њ
yЊ
Equilateral triangles are also equiangular.

Example 27. 25Њ 28.
yЊ
What is mjG? hsm1y1Њ gmxse_04cr_t05093

Since EF ≅ EG, ∠F ≅ ∠G by the hsm11gmse_04cr_t05092
Isosceles Triangle Theorem. So xЊ 25Њ
m∠G = 30. E 7
30Њ
F G

hsm11gmse_04cr_t05094hsm11gmse_04cr_t05095

hsm11gmse_04cr_t05091

Chapter 4  Chapter Review 275

4-6  Congruence in Right Triangles

Quick Review Exercises

If the hypotenuse and a leg of one right triangle are Write a proof for each of the following.
congruent to the hypotenuse and a leg of another 29. Given:  LN # KM, KL ≅ ML
right triangle, then the triangles are congruent by the Prove:  △KLN ≅ △MLN
Hypotenuse-Leg (HL) Theorem.
L
Example
K NM
Which two triangles are congruent? Explain. 30. Given:  PS # SQ, RQ # QS,

BM N PQ ≅ RS

AL Y Prove:  △hsPmSQ11≅g△mRsQe_S04cr_t0509Q7 P
CZ S
R
X
Since △ABC and △XYZ are right triangles with congruent
legs, and BC ≅ YZ, △ABC ≅ △XYZ by HL.

hsm11gmse_04cr_t05096

4-7  Congruence in Overlapping Triangles hsm11gmse_04cr_t05098

Quick Review Exercises

To prove overlapping triangles congruent, you look for the Name a pair of overlapping congruent triangles in each
common or shared sides and angles. diagram. State whether the triangles are congruent by

SSS, SAS, ASA, AAS, or HL.

Example E F 31. A 32. F G

Separate and redraw the

overlapping triangles.

Label the vertices. DC

ED CB I H

E FE F
33. P Q R

AS

D hsm1C1gmse_04cr_t05099 hsm11gmsTe_04cr_t05101hsm11gmse_04cr_t05102

hsm11gmse_04cr_t05100 hsm11gmse_04cr_t05103

276 Chapter 4  Chapter Review

4 Chapter Test MathX

OLMathXL® for School
R SCHO Go to PowerGeometry.com
L®

FO

Do you know HOW? 8. △CEO ≅ △HDF . Name all of the pairs of
Write a congruence statement for each pair of triangles. corresponding congruent parts.

1. P L 9. Algebra  Find the value of x.

YA 108Њ
3xЊ

2. N S

O

hsm11gmse_04ct_t02574 Name a pair of overlapping congruent triangles in each

diagram. State whether the triangles are congruent by

E SSS, SAS, ASA,hAsAmS,1o1rgHmL.se_04ct_t02582
C D
10. Given:  CE ≅ DF ,
Which postulate or theorem, if any, could you use
to prove the two triangles congruent? If not enough CF ≅ DE

informhsamtio1n1igs mgivseen_,0w4rcitte_nt0o2t e5n7o6ugh information. FE

3. 11. Given:  RT ≅ QT , QR
AT ≅ ST AS

hsm11gmTse_04ct_t02585

4. Do you UNDERSTAND?

hsm11gmse_04ct_t02577 12. Reasoning  Isosceles △ABC, with right ∠B, has

5. a point D on AC such △thAatBBDDan#dhA△sCmC.BW1D1h?gaEtmxispstleha_ien0.4ct_t02586
relationship between
hsm11gmse_04ct_t02578
Write a proof for each of the following.
6.
13. Given:  AT ≅ GS, G
hsm11gmse_04ct_t02579 AT } GS
A
7. Prove:  △GAT ≅ △TSG
S
hsm11gmse_04ct_t02580
14. Given:  LN bisects ∠OLM T
and ∠ONM. L

Prove:  ON ≅ MN hsm11gmse_04ct_t02583

OM

N

Chapter 4  Chapter Test 277

4 Common Core Cumulative ASSESSMENT
Standards Review

Some test questions ask you In the coordinate plane, the vertices of TIP 1
to compare geometric figures. △ABC are A( -2, 5), B(2, 5), and C( -2, 2).
Read the sample question at Which of the following are the side lengths Find the lengths of the sides
the right. Then follow the tips to for △DEF , such that△ABC ≅ △DEF ? of △ABC.
answer it.
DE = 3, DF = 4, EF = 4 Think It Through
TIP 2
DE = 3, DF = 4, EF = 5 Vertices A and B have the same
You can use SSS to show y‑coordinate and vertices A and C
triangles are congruent. DE = 4, DF = 3, EF = 5 have the same x-coordinate, so
Find the corresponding
side lengths of  △DEF. DE = 4, DF = 5, EF = 5 AB = 0 -2 - 2 0 = 0 -4 0 = 4 and
AC = 0 5 - 2 0 = 0 3 0 = 3.

By the Distance Formula,

BC = 2(4)2 + (3)2 = 125 = 5.
So DE = AB = 4, DF = AC = 3,
and EF = BC = 5. The correct
answer is C.

VLVeooscsacoabnubluarlayry Builder Selected Response

As you solve test items, you must Read each question. Then write the letter of the correct
understand the meanings of mathematical answer on your paper.
terms. Match each term with its
mathematical meaning. 1. Given:  DE } CB, A
∠ADE ≅ ∠AED

A. slope I. lines that intersect to form Prove:  AC ≅ AB

B . perpendicular right angles Proof:  Since DE } CB,
∠ACB ≅ ∠ADE and
lines II. having the same size and D E

C . polygon shape ∠AED ≅ ∠ABC by

D. conjecture III. a conclusion reached by the Corresponding C B
inductive reasoning Angles Theorem. Since

E. congruent IV. a closed plane figure with ∠ADE ≅ ∠AED, ∠ACB ≅ ∠ABC by the

at least three sides that are Transitive Property. Which theorem or

segments definition proves that AC ≅ AB?

Isosceles Triangle Theorem hsm11gmse_04cu_t05070

V. the ratio of the vertical Converse of Isosceles Triangle Theorem
change (rise) to the
horizontal change (run) Alternate Interior Angles Theorem

Definition of congruent segments

278 Chapter 4  Common Core Cumulative Standards Review

2. Which statement must be true for two polygons to be 6. Which condition allows you to prove that / } m?
congruent?
18 ᐉ
All the corresponding sides should be congruent. 23
All the corresponding sides and angles should
45 m
be congruent.
All the corresponding angles should be congruent.
All sides in each polygon should be congruent.
∠1 ≅ ∠8 ∠3 ≅ ∠4
3. If △ABC ≅ △CDA, which of the following must ∠2 ≅ ∠8 ∠3 ≅ ∠5
be true?
7. A line pashssems th1r1ogumghs(e3_, 0-44c)ua_ndt0h5a0s7a2slope of -5.
AB
How can you find the y-intercept of the graph?
D C
Substitute -5 for b, -4 for x, and 3 for y in
AB ≅ CA ∠CAB ≅ ∠ACD y = mx + b. Then solve for m, the y-intercept.

BC ≅ DC ∠ABC ≅ ∠CAD Substitute -5 for b, 3 for x, and -4 for y in
y = mx + b. Then solve for m, the y-intercept.
4. Given:  ∠1 ≅hs∠m21, A1Bgm≅sAeC_04cu_t05069
Substitute -5 for m, -4 for x, and 3 for y in
What additional information do you need to prove y = mx + b. Then solve for b, the y-intercept.
△ABD ≅ △ACE by AAS?
Substitute -5 for m, 3 for x, and -4 for y in
y = mx + b. Then solve for b, the y-intercept.

A 8. Given:  △RST ≅ △LMN
Which reason could you use to prove that ∠R ≅ ∠L?
15 3 4 62 SAS
BD EC
SSS
AD ≅ AE ∠5 ≅ ∠6
BD ≅ EC BE ≅ DC ASA

Corresponding parts of congruent triangles
are congruent.

9. Which equation represents the perpendicular bisector

5. Which of thhesfmol1lo1wginmgsseta_t0em4ceun_tsti0s5tr0u7e1? of the segment shown?

Point, line, and plane are undefined terms. y=x-3 y
A theorem is an accepted statement of fact.
“Vertical angles are congruent” is a definition. 3x + 3y = 3 4
A postulate is a conjecture that is proven. y = 3x (5, 2)
x + y = 3
2

x
O 46

Ϫ2 (1, ؊2)

Ϫ4

hsm11gmse_04cu_t05073
Chapter 4  Common Core Cumulative Standards Review 279

Constructed Response 16. Describe how the following everyday meanings
of acute and obtuse help you to remember their
1 0. One bag of garden soil covers approximately 16 ft2. If mathematical meanings.
it takes 16p ft of fencing to enclose a circular garden,
how many bags of soil do you need to cover acute  adj.  Having a sharp point
the garden?
obtuse  adj.  Not sharp or pointed; blunt
11. What is the value of x in the figure below?
17. Write a proof for the following.
D
Given:  AE ≅ DE, EB ≅ EC
75Њ Prove:  △AEB ≅ △DEC

A D

44Њ (5x Ϫ 6)Њ E
E BC
F

1 2. The length of a rectangle is seven more than three Extended Response
times its width. If the perimeter is 38 cm, what is the
area, in square centimeters, of the rectangle? 18. hResamd1th1igsmexsceer_p0t4frcoum_ta0n5o0n7li8ne news article.

13. What is thhesvmal1u1e gofmxsine_th0e4fciguu_rte0b5e0lo7w4?

(2x Ϫ 5)Њ

135Њ Halley’s Comet can be seen periodically at its perihelion, the
shortest distance from the sun during its orbit. Mark Twain was
1 4. ABCD ≅ WXYZ. What is WX? born two weeks after the comet’s perihelion. In his
biography he said, “I came in with Halley’s Comet in 1835. It is
A 3 B Y coming again next year, and I expect to go out with it.” Twain
died in 1910, the day after the comet’s perihelion. The most
2 hs4m11gmse_04cu_t0Z5075 recent sighting of Halley’s Comet was in 1986. Its next
D appearance is expected in 2061.
2
4 a. Make a conjecture about the year in which
CX Halley’s Comet will appear after 2061. Explain
W your reasoning.

1 5. Amy is designing a ramp up to a 16-in.-high hsmb1. 1Hgomwsceo_n0fi4decnut_atr0e5y0o7u9about your conjecture?

skateboarding platform, as shown on the graph below. Explain.
1
shIfhssomhuel1dw1saghnmetscsthehoe_o0ssl4eocpfouer_otthf0teh5xe0-cr7ao6morpditnoabtee 3 , what value 19. The coordinates of the vertices of rectangle LMNK
the top of are L( -2, 5), M(2, 5), N(2, 3), and K( -2, 3). The
at coordinates of the vertices of rectangle PQRS are
P(3, 0), Q(3, -3), R(1, -3), and S(1, 0). Are these two
the ramp? rectangles congruent? Explain why or why not. If
not, how could you change the vertices of one of the
y (x, 16) rectangles to make them congruent?

(60, 16)

x
O (60, 0)

hsm11gmse_04cu_t05077

280 Chapter 4  Common Core Cumulative Standards Review

Get Ready! CHAPTER

Lesson 1-6 Basic Constructions 5

Use a compass and straightedge for each construction.

1. Construct the perpendicular bisector of a segment.

2. Construct the bisector of an angle.

Lesson 1-7 The Midpoint Formula and Distance Formula

Find the coordinates of the midpoints of the sides of △ABC. Then find the
lengths of the three sides of the triangle.

3. A(5, 1), B( -3, 3), C(1, -7)

4. A( -1, 2), B(9, 2), C( -1, 8)

5. A( -2, -3), B(2, -3), C(0, 3)

Lesson 2-2 Finding the Negation

Write the negation of each statement.

6. The team won. 7. It is not too late. 8. m∠R 7 60

Lesson 3-7 Slope

Find the slope of the line passing through the given points.

9. A(9, 6), B(8, 12) 10. C(3, -2), D(0, 6) 11. E( -3, 7), F( -3, 12)

Looking Ahead Vocabulary

12. The altitude of an airplane is the height of the airplane above ground. What do you
think an altitude of a triangle is?

13. The distance between your home and your school is the length of the shortest path
connecting them. How might you define the distance between a point and a line in
geometry?

14. In Chapter 1, you learned the definition of a midpoint of a segment. What do you
think a midsegment of a triangle is?

15. If two parties are happening at the same time, they are concurrent. What would it
mean for three lines to be concurrent?

Chapter 5  Relationships Within Triangles 281

CHAPTER Relationships Within

5 Triangles

Download videos VIDEO Chapter Preview 1 Coordinate Geometry
connecting math Essential Question  How do you use
to your world.. 5-1 Midsegments of Triangles coordinate geometry to find relationships
5-2 Perpendicular and Angle Bisectors within triangles?
Interactive! ICYNAM 5-3 Bisectors in Triangles
Vary numbers, ACT I V I TI 5-4 Medians and Altitudes 2 Measurement
graphs, and figures D 5-5 Indirect Proof Essential Question  How do you solve
to explore math ES 5-6 Inequalities in One Triangle problems that involve measurements of
concepts.. 5-7 Inequalities in Two Triangles triangles?

The online 3 Reasoning and Proof
Solve It will get Essential Question  How do you write
you in gear for indirect proofs?
each lesson.

Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • Congruence
Spanish English/Spanish Vocabulary Audio Online:
• Similarity, Right Triangles, and Trigonometry
English Spanish
• Mathematical Practice: Construct viable
altitude of a triangle, p. 310 altura de un triángulo arguments

Online access centroid, p. 309 centroide
to stepped-out
problems aligned circumcenter, p. 301 circuncentro
to Common Core
Get and view concurrent, p. 301 concurrente
your assignments
online. NLINE equidistant, p. 292 equidistante
ME WO
O incenter, p. 303 incentro
RK
HO indirect proof, p. 317 prueba indirecta

median, p. 309 mediana

Extra practice midsegment of a segmento medio de
and review triangle, p. 285 un triángulo
online
orthocenter, p. 311 ortocentro

Virtual NerdTM
tutorials with
built-in support

PERFORMANCE TASK

Common Core Performance Task

Estimating the Length of a Hiking Trail

Several hikers plan to hike along some trails from their campground to a waterfall.
The trail map below shows their hike with a dashed line. Unfortunately, the lengths
of some of the trails are missing on this map. The hikers would like to find a range of
values for the total length of their hike.

A 4.2 km

2.5 km D
3.1 km
Campground L
C

3.5 km 4.8 km Waterfall
W
B 2.2 km K 5.1 km
3.1 km 4.8 km

5.1 km 56° 50°37° 5.1 km 3.2 km
N 5.1 km X
G

Task Description

Find a range, in kilometers, for the length of the group’s hike from the
campground to the waterfall.

Connecting the Task to the Math Practices MATHEMATICAL

As you complete the task, you’ll apply several Standards for Mathematical PRACTICES
Practice.

• You’ll analyze the relationships created by a segment that joins the midpoints
of two sides of a triangle. (MP 7)

• You’ll use reasoning to apply inequalities in a triangle. (MP 3)

• You’ll interpret your results when you apply inequalities in pairs of triangles. (MP 2)

Chapter 5  Relationships Within Triangles 283

Concept Byte Investigating MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Midsegments
Use With Lesson 5-1 Prepares for MG-ACFOS.C.9.120. GP-rCovOe.3th.1e0o rePmrosvaebtohuetorems
atrbiaonugtltersia.n.g.les . . .
TECHNOLOGY
MP 5

A
Midsegment
Step 1 Use geometry software to draw and label △ABC. Construct the midpoints E
D and E of AB and AC. Connect the midpoints with a midsegment.
DC
Step 2 Measure DE and BC. Calculate DBCE.
Step 3 Measure the slopes of DE and BC. B

Step 4 Manipulate the triangle and observe the lengths and slopes of DE and BC.

Exercises hsm11gmse_0501a_t06184.ai

1. Make a Conjecture  Make conjectures about the lengths and slopes A
of midsegments. E

2. Construct the midpoint F of BC. Then construct the other two midsegments of  DC
△ABC. Test whether these midsegments support your conjectures in Exercise 1. F

3. △ABC and the three midsegments form four small triangles. B
a. Measure the sides of the four small triangles and list those that you find

are congruent.
b. Make a Conjecture  Make a conjecture about the four small triangles formed

by a triangle and its three midsegments.

For the remaining exercises, assume your conjectures in Exercises 1 and 3 are true. hsm11gmse_0501a_t06185.ai

4. What can you say about the areas of the four small triangles in the A
window above?
GE C
5. a. How does △ABC compare to each small triangle in area? DH
b. How does △ABC compare to each small triangle in perimeter?
I
6. Construct the three midsegments of △DEF . Label this triangle △GHI . F
a. How does △ABC compare to △GHI in area?
b. How does △ABC compare to △GHI in perimeter? B
c. Suppose you construct the midsegment triangle inside △GHI . How would

△ABC compare to this third midsegment triangle in area and perimeter?

hsm11gmse_0501a_t06186.ai

284 Concept Byte  Investigating Midsegments

5-1 Midsegments of MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Triangles
GM-ACFOS.C.9.120. GP-rCovOe.3th.1e0o rePmrosvaebtohuetotrreimansgalebso.u.t .
trhiea nsgelgems e. n. .t tjhoein sinegmtheenmt jiodipnoiningttshoefmtwidopsoiidnets of tawo
stridiaensgolef aistpriaarnagllleel itsoptahreatllheilrdtostidhe athnidrdhsaildfethaendlenhgaltfht.h.e.
lAelnsgothG.-.C.OA.lDs.o12M, AGF-SR.9T1.B2.5G-CO.4.12, G-SRT.2.5

MP 1, MP 3, MP 4, MP 5

Objective To use properties of midsegments to solve problems

Check with your Cut out a triangle of any shape. Label L C
classmates. Do its largest angle C, and the other N
they get the same angles A and B. Fold A onto C to find A
results? the midpoint of AC. Do the same for L DB
BC. Label the midpoints L and N, and N
MATHEMATICAL then draw LN. A
L DB
PRACTICES Fold the triangle on LN as shown. N
M
Fold A to D and fold B to D. DP B

Label the vertices M and P as shown.
What is the relationship between
MP and AB? How do you know?
What conjecture can you make about
the relationship between LN and AB?

Lesson In the Solve It, LN is a midsegment of △ABC. A midsegment of a triangle is a segment
connecting the midpoints of two sides of the triangle.
Vocabulary
• midsegment of a Essential Understanding  There are two special relationships between a
midsegment of a triangle and the third side of the triangle.
triangle

Theorem 5-1  Triangle Midsegment Theorem

Theorem If . . . Then . . .
If a segment joins the
midpoints of two sides of a D is the midpoint of CA and DE } AB and
triangle, then the segment 1
is parallel to the third side E is the midpoint of CB DE = 2 AB
and is half as long. C

D E
A B

You will prove Theorem 5-1 in Lesson 6-9.

hsm11gmse_0501_t06128.ai 285

Lesson 5-1  Midsegments of Triangles

Here’s Why It Works  You can verify that the Triangle y
6 A(4, 6)
Midsegment Theorem works for a particular triangle. Use the
1
following steps to show that DE } AB and that DE = 2 AB for

a triangle with vertices at A(4, 6), B(6, 0), and C(0, 0), where 4D
2
D and E are the midpoints of CA and CB.
CE
( )Step 1 x1 + x2 , y1 + y2
Use the Midpoint Formula, M = 2 2 , x
B(6, 0)
to find the coordinates of D and E.

( ) 0 + 4 0 + 6 = D(2, 3).
The midpoint of CA is D 2 , 2

( ) 0 + 6, 0 + 0 = E(3, 0).
The midpoint of CB is E 2 2

Step 2 To show that the midsegment DE is parallel to the side ABh, sfimnd1t1hgemslospee_,0501_t06127.ai
xy22 - xy11 ,
m = - of each segment.

slope of DE = 0- 32 slope of AB = 0 -6
3- 6 -4

= -13 = -6
2

= -3 = -3

Step 3 To show DE = 1 AB, use the Distance Formula, d = 2(x2 - x1)2 + (y2 - y1)2
2
to find DE and AB.

DE = 2(3 - 2)2 + (0 - 3)2 AB = 2(6 - 4)2 + (0 - 6)2

= 11 + 9 = 14 + 36

= 110 = 140

= 2110

Since 110 = 1 (2 110 ), you know that DE = 1 AB.
2 2

Problem 1 Identifying Parallel Segments E

How do you identify What are the three pairs of parallel segments in △DEF ? R S
a midsegment? RS, ST , and TR are the midsegments of △DEF. By the Triangle DT F

Look for indications Midsegment Theorem, RS } DF , ST } ED, and TR } FE.
that the endpoints

of a segment are the N

midpoints of a side of Got It? 1. a. I n △XYZ, A is the midpoint of XY, B is the 65Њ hsm11Ugmse_0501_t06129.ai

the triangle. midpoint of YZ, and C is the midpoint of ZX .

What are the three pairs of parallel segments?

b. R easoning  What is m∠VUO in the figure at the

right? Explain your reasoning. O

MV

286 Chapter 5  Relationships Within Triangles hsm11gmse_0501_t13162

Problem 2 Finding Lengths

In △QRS, T, U, and B are midpoints. What are the lengths of QUR
TU , UB, and QR?
Which relationship
stated in the Triangle Use the relationship T 30 B 40
Midsegment Theorem 50
should you use? length of a midsegment = 1 (length of the third side)
You are asked to find 2
lengths, so use the to write an equation about the length of each midsegment.
relationship that refers
to the lengths of a TU = 1 SR UB = 1 QS TB = 1 QR S
midsegment and the 2 2 2
third side.
= 1 (40) = 1 (50) 30 = 1 QR
2 2 2

= 20 = 25 60 = QR ADC

Got It? 2. In the figure at the right, AD = 6 and DE = 7.5. hsm11gFmse_0501_tE06135.ai
What are the lengths of DC, AC, EF , and AB?
B

You can use the Triangle Midsegment Theorem to find lengths hsm11gmse_0501_t06138.ai
of segments that might be difficult to measure directly.

Problem 3 Using a Midsegment of a Triangle STEM

Why does the Environmental Science  A geologist wants to determine
geologist find the the distance, AB, across a sinkhole. Choosing a point E
length of CD? outside the sinkhole, she finds the distances AE
and BE. She locates the midpoints C and D of
CD is a midsegment of
△AEB, so the geologist AE and BE and then measures CD. What
can use its length to find is the distance across the sinkhole?
AB, the distance across
the sinkhole. CD is a midsegment of △AEB.

CD = 1 AB △ Midsegment Thm.
2
Substitute 46 for CD.
  46 = 1 AB Multiply each side by 2.
2

  92 = AB

The distance across the sinkhole is 92 ft.

B
D

E 46 ft

C 963 ft Bridge
A 2640 ft
C
Got It? 3. CD is a bridge being built over a lake, 963 ft D
as shown in the figure at the right.
What is the length of the bridge?

Lesson 5-1  Midsegments of Triangles 287

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Use the figure at the right for Exercises 1–3.
PRACTICES

4. Vocabulary  How does the term midsegment describe

1. Which segment is parallel L the segments discussed in this lesson?
to JK ?
5. Reasoning  If two noncollinear segments in the

2. If LK = 46, what is NM? coordinate plane have slope 3, what can you conclude?

3. If JK = 5x + 20 and N O 6. Error Analysis  A student sees O
NO = 20, what is the value J M
of x? this figure and concludes that L

K PL } NO. What is the error in N P T
the student’s reasoning?

hsm11gmse_0501_t06192.ai MATHEMATICAL hsm11gmse_0501_t06141.ai
Practice and Problem-Solving Exercises
PRACTICES See Problem 1.

A Practice Identify three pairs of parallel segments in each diagram.

7. V 8. H
U W
5 6
T YX G J
5 6

F 7L7 K

Name the segment that is parallel to the A

givenhssemgm11egnmt. se_0501_t06144.ai F hsm11gmG se_0501_t06147.ai
9. AB 10. BC
C EB
11. EF 12. CA

13. GE 14. FG

Points E, D, and H are the midpoints of the sides hsm1T1gmse_0501_t06190.ai See Problem 2.
of △TUV. UV = 80, TV = 100, and HD = 80.
HE
15. Find HE. 16. Find ED. V DU

17. Find TU. 18. Find TE. 21.

Algebra  Find the value of x. 20.
19.

x 5x hsm11gmse_0501_t06193.ai
45
72

26 6x

hsm11gmse_0501_t06194.ai hsm11gmse_0501_t06195.ai

288 Chapter 5  Relationships Within Triangles hsm11gmse_0501_t06196.ai

Algebra  Find the value of x.

22. 17 23. 38 24.
xϪ4 5x Ϫ 4
xϩ2

8

25. Surveying  A surveyor needs to measure the distance PQ P See Problem 3.
N
across the lake. Beginning at point S, shhsemlo1c1atgems thsee_0501_t06198.ai
mhsidmp1oi1ngtsmosf eSQ_0a5n0d1S_Pt0a6t M19a7n.daiN. She then measures NM. hsm11gmse_0501_t061S99.ai

What is PQ? M 78 m

B Apply 26. Kayaking  You want to paddle your kayak across a lake. To Q

determine how far you must paddle, you pace out a triangle, 80 150

counting the number of strides, as shown.

a. If your strides average 3.5 ft, what is the length of the 80 hsm11gmse_0501510_t13164

longest side of the triangle? 250
b. What distance must you paddle across the lake?

27. Architecture  The triangular face of the Rock and

Roll Hall of Fame in Cleveland, Ohio, is isosceles. The

length of the base is 229 ft 6 in. Each leg is divided into hsm11gmse_0501_t06191.ai

four congruents parts by the red segments. What is the

length of the white segment? Explain your reasoning.

28. Think About a Plan  Draw △ABC. Construct another
triangle so that the three sides of △ABC are the

midsegments of the new triangle.
• Can you visualize or sketch the final figure?
• Which segments in your final construction will be

parallel?

29. Writing  In the figure at the right, m∠QST = 40. What is Q
m∠QPR? Explain how you know.

30. Coordinate Geometry  The coordinates of the vertices of a S T
triangle are E(1, 2), F(5, 6), and G(3, -2). PU R

a. Find the coordinates of H, the midpoint of EG, and J,

the midpoint of FG.
bc.. SShhooww tthhaatt HHJJ }=E12FE.F .

X is the midpoint of UV. Y is the midpoint of UW. hsm11gmse_0501_t06148.ai

31. If m∠UXY = 60, find m∠V. U Y
W
32. If m∠W = 45, find m∠UYX . X
33. If XY = 50, find VW.

34. If VW = 110, find XY. V

Lesson 5-1  Midsegments of Triangles 289

hsm11gmse_0501_t06200.ai

IJ is a midsegment of △FGH. IJ = 7, FH = 10, and H J
GH = 13. Find the perimeter of each triangle. I G
F
35. △IJH

36. △FGH

37. Kite Design  You design a kite to look like the one at the right.

Its diagonals measure 64 cm and 90 cm. You plan to use

ribbon, represented by the purple rectangle, to connect thhsem11gmse_0501_t06201.ai
midpoints of its sides. How much ribbon do you need?

77 cm 154 cm

122 cm 308 cm

Algebra  Find the value of each variable. 39.
38. x
21
30

x 25

40. 41. 3x Ϫ 6

hsmx 11g5mse_0501_t06202.ai hsm1y 1gmse_0501_t06203.ai

60Њ x 2x ϩ 1

Use the figure at the right for Exercises 42–44. D

42. DhFsm= 2141,gBmC s=e6_,0a5n0d1D_Bt0=6280. F4i.nadi the perimeter hsm11gB mse_05C01_t06205.ai

of △ADF .
43. Algebra  If BE = 2x + 6 and DF = 5x + 9, find DF.

44. Algebra  If EC = 3x - 1 and AD = 5x + 7, find EC. AE F

45. Open-Ended  Explain how you could use the Triangle Midsegment Theorem as the

basis for this construction: Draw CD. Draw point A not on ChDs.mCo1n1sgtrmucsteA_B0s5o01_t06206.ai
1
that AB } CD and AB = 2 CD.

C Challenge 46. Reasoning  In the diagram at the right, K, L, and M are the midpoints of the B

sides of △ABC. The vertices of the three small purple triangles are the

midpoints of the sides of △KBL, △AKM, and △MLC. The perimeter of

△ABC is 24 cm. What is the perimeter of the shaded region? K L

AMC

290 Chapter 5  Relationships Within Triangles hsm11gmse_0501_t06208.ai

47. Coordinate Geometry  In △GHJ, K(2, 3) is the midpoint of GH, L(4, 1) is

the midpoint of HJ, and M(6, 2) is the midpoint of GJ . Find the coordinates
of G, H, and J.

48. Complete the Prove statement and then write a proof. Y
Proo f Given:  In △VYZ, S, T, and U are midpoints. ST

Prove:  △YST ≅ △TUZ ≅ △SVU ≅ ?

VU Z

PERFORMANCE TASK hsm11gmsPeMR_ATA0H5CEM0TAI1TCI_CEAtSL06207.ai

Apply What You’ve Learned MP 7

Look at the trail map from page 283, shown again below.

A 4.2 km

2.5 km D L
Campground 3.1 km

C

3.5 km 4.8 km Waterfall
3.1 km W
B 2.2 km K 5.1 km

4.8 km 5.1 km 5.1 km 3.2 km

G 56° 50°37° 5.1 km X
N

Select all of the following that are true. Explain your reasoning.
A. In △DGL, BL is a midsegment.
B. In △DGL, BK is a midsegment.
C. BK is parallel to DL.
D. BL is parallel to GK.
E. The length of BK is half the length of BL.
F. The length of DL is twice the length of BK.
G. DL is the shortest side of △DGL.

Lesson 5-1  Midsegments of Triangles 291

5-2 Perpendicular and MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Angle Bisectors
MG-ACFOS.C.9.192  .PGro-vCeOt.h3e.o9r ePmrsovaebothuet olirneems sanadboauntgleinse.s. .
aponidntasnognleas p. e. r.ppeonidnitcsuolanr abispeecrtpoernodficaullianrebsiesgemcteonrtoafre
aexlainctelysethgomsenetqaurideisetxaancttflyrotmhotshee esqegumideisntta’sntenfrdopmointhtse.
sAelgsmoeGn-tS’sReTn.dBp.5oints. Also MAFS.912.G-SRT.2.5

MP 1, MP 3, MP 4, MP 5, MP 8

Objective To use properties of perpendicular bisectors and angle bisectors

Confused? Try You hang a bulletin board over your
drawing a diagram desk using string. The bulletin board
to “straighten” is crooked. When you straighten the
yourself out. bulletin board, what type of triangle
does the string form with the top
MATHEMATICAL of the board? How do you know?
Visualize the vertical line along the wall
PRACTICES that passes through the nail. What
relationships exist between this line
and the top edge of the straightened
bulletin board? Explain.

Lesson In the Solve It, you thought about the relationships that must exist in order for a bulletin
board to hang straight. You will explore these relationships in this lesson.
Vocabulary
• equidistant Essential Understanding  There is a special relationship between the points on
• distance from a the perpendicular bisector of a segment and the endpoints of the segment.
In the diagram below on the left, <CD> is the perpendicular bisector of AB. <CD> is
point to a line perpendicular to AB at its midpoint. In the diagram on the right, CA and CB are drawn
to complete △CAD and △CBD.

CC

AD B AD B

You should recognize from your work in Chapter 4 that △CAD ≅ △CBD. So you can
conclude that CA ≅ CB, or that CA = CB. A point is equidistant from two objects if it is

the same distance fromhstmhe1o1bgjemctss.eS_o0p5o0i2nt_Ct0i6s 0eq7u1idistant from points A and B.

This suggests a proof of Theorem 5-2, the Perpendicular Bisector Theorem. Its converse
is also true and is stated as Theorem 5-3.

292 Chapter 5  Relationships Within Triangles

Theorem 5-2  Perpendicular Bisector Theorem

Theorem I<PfM. >. . AB and MA = MB Then . . .
If a point is on the # PA = PB
perpendicular bisector
of a segment, then it is P P
equidistant from the
endpoints of the segment.

AM B AM B

You will prove Theorem 5-2 in Exercise 32.

Theorem 5-3  Converse of the Perpendicular Bisector Theorem

Theorem Ifh.s.m. 11gmse_0502_t06072 T<PhhMse>mn#1. A.1.BgmansdeM_0A5=02M_Bt06074
If a point is equidistant
from the endpoints of a PA = PB
segment, then it is on the
perpendicular bisector of PP
the segment.

A B AM B

You will prove Theorem 5-3 in Exercise 33.

hsm11gmse_0502_t06081 hsm11gmse_0502_t06082
Problem 1 Using the Perpendicular Bisector Theorem

How do you know BD Algebra  What is the length of AB? A
is the perpendicular
bisector of AC ? BD is the perpendicular bisector of AC, so B is equidistant from 4x D
The markings in the A and C.
d iagram show that BD is
pt heerpmeniddpicouinlat rotfoAACC. at BA = BC Perpendicular Bisector Theorem

4x = 6x - 10 Substitute 4x for BA and 6x - 10 for BC. B 6x Ϫ 10 C

- 2x = - 10 Subtract 6x from each side.

x=5 Divide each side by - 2.

Now find AB. hsm11gmse_0502_t06083
AB = 4x
AB = 4(5) = 20 Substitute 5 for x.

Got It? 1. What is the length of QR? P S R

3n Ϫ 1 5n Ϫ 7

Q

Lesson 5-2h smPe1rp1egnmdicsuel_ar0a5n0d2A_tn0g6le0B8i5sectors 293

How do you find Problem 2 Using a Perpendicular Bisector
points that are
equidistant from two A park director wants to build a T-shirt stand equidistant from the Rollin’ Coaster
given points? and the Spaceship Shoot. What are the possible locations of the stand? Explain.
By the Converse of the
Perpendicular Bisector Paddle boats S
Theorem, points P Spaceship Shoot
equidistant from two
given points are on the Rollin’ Coaster M
perpendicular bisector of R Merry-go-round
the segment that joins
the two points.

To be equidistant from the two rides, the stand P ᐍ S
R A M
should be on the perpendicular bisector of the

segment connecting the rides. Find the midpoint A
of RS and draw line / through A perpendicular to
RS. The possible locations of the stand are all the
points on line /.

Got It? 2. a. Suppose the director wants the T-shirt stand to be equidistant from the
paddle boats and the Spaceship Shoot. What are the possible locations?

b. Reasoning  Can you place the T-shirt sthansdms1o1tghamt isteis_e0q5u0id2i_stta1n3t1fr6o5m

the paddle boats, the Spaceship Shoot, and the Rollin’ Coaster? Explain.

Essential Understanding  There is a special relationship between the points
on the bisector of an angle and the sides of the angle.

The distance from a point to a line is the length of the A

perpendicular segment from the point to the line. This distance

is also the length of the shortest segment from the point to the

line. You will prove this in Lesson 5-6. In the figure at the right, Bᐉ
the distances from A to / and from B to / are represented by the

red segments.

In the diagram, AD> is the bisector of ∠CAB. If you measure the 0123456
hsm11gmse_0C502_t06088
lengths of the perpendicular segments from D to the two sides of

the angle, you will find that the lengths are equal. Point D is equidistant

from the sides of the angle.

D
AB

294 Chapter 5  Relationships Within Triangles

Theorem 5-4  Angle Bisector Theorem

Theorem aIQfnS.d> .bS.iRse#ctsQ∠RP> QR, SP # QP>, Then . . .
If a point is on the bisector P SP = SR
of an angle, then the
point is equidistant from QS P
the sides of the angle. QS

RR

You will prove Theorem 5-4 in Exercise 34.

Theorem 5-5  Converse of the Angle Bisector Theorem

Theorem IShfPs.m#. .1Q1Pg>,mSRse#_0Q5R0>,2_t06090 TQhhSse>mbni1s.e1.cg.tsm∠sPeQ_0R502_t06107
If a point in the interior

of an angle is equidistant and SP = SR

from the sides of the P P
angle, then the point is on

the angle bisector. Q S QS

RR

You will prove Theorem 5-5 in Exercise 35.

Problem 3 Using the Anhgsmle1B1igsemcstoer_0T5h0e2o_rte0m6106 hsm11gmse_0502_t06107

Algebra  What is the length of RM? 2x ϩ 25

NR> b#iseNcLt>sa∠ndLNRQP.# NQ>. The length Use the Angle Bisector L 7x Q
RM of RM Theorem to write an equation MR P
you can solve for x.

RM = RP Angle Bisector Theorem N

7x = 2x + 25 Substitute. hsm11gmse_0502_t06092

H ow can you use the 5x = 25 Subtract 2x from each side. B
ef oxrpRrePstsoiocnhgeicvkeynour C F 6x ϩ 3
answer? x=5 Divide each side by 5.
eS xupbrsetistsuioten52fxo+r x2i5n the 4x ϩ 9
a nd verify that the result Now find RM. D
is 35.
RM = 7x

= 7(5) = 35 Substitute 5 for x.

Got It? 3. What is the length of FB?

Lesson 5-2  Perpendicular and Angle Bisectors 295

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Use the figure at the right for Exercises 1–3.
PRACTICES

4. Vocabulary  Draw a line and a point not on the line.

1. What is the relationship D Draw the segment that represents the distance from
between AC and BD?
15 E the point to the line.
2. What is the length of AB? A 18
C 5. Writing  Point P is in the interior of ∠LOX . Describe
3. What is the length of DC? B
how you can determine whether P is on the bisector
of ∠LOX without drawing the angle bisector.

Practice and Problehmsm-S1o1glvmisneg_05E0x2e_rt0c6is10e3s MATHEMATICAL

PRACTICES

A Practice Use the figure at the right for Exercises 6–8. M See Problem 1.

6. What is the relationship between MB and JK ? 9x Ϫ 18 3x
7. What is value of x?
8. Find JM. J BK

Reading Maps  For Exercises 9 and 10, use the map of a part of Manhattan. See Problem 2.

9. Which school is equidistant from the subway hsm11gmse_0502_t06094M = Subway Station
stations at Union Square and 14th Street? How do 8th Ave
7th Ave
you know? 6th Ave
Broadway
10. Is St. Vincent’s Hospital equidistant from Village NYC Museum
Kids Nursery School and Legacy School? How do School
you know?
M14th St Village Kids Coleman School
11. Writing  On a piece of paper, mark a point H for Nursery School
home and a point S for school. Describe how to find Xavier School
the set of points equidistant from H and S. St. Vincent’s
Hospital Legacy School
Union Square
M

Use the figure at the right for Exercises 12–15. See Problem 3.
12. AHcKc>o? rFdrionmg tHoFth>?e diagram, how far is L from
13. How is HL> related to ∠KHF ? Explain. L
14. Find the value of y. 27
15. Find m∠KHL and m∠FHL.
KF
6yЊ H (4y ϩ 18)Њ

hsm11gmse_0502_t06116

296 Chapter 5  Relationships Within Triangles

16. Algebra Find x, JK, and JM. 17. Algebra Find y, ST, and TU.

K 5y T
xϩ5 S

JL 3y ϩ 6

2x Ϫ 7 VU
M

B Apply Algebra  Use the figure at the right for Exercises 18–22.

18. Find the value of x. T

19. FhinsdmT1W1.gmse_0502_t06111 hsm11gmse_0502_t06114

20. Find WZ. Y

2x

21. What kind of triangle is △TWZ? Explain. W 3x Ϫ 5 Z

22. If R is on the perpendicular bisector of TZ, then
R is ? from T and Z, or ? = ? .

23. Think About a Plan  In the diagram at hsm11gmse_0502_t06117.ai L
the right, the soccer goalie will prepare XG
for a shot from the player at point P by
moving out to a point on XY . To have P
the best chance of stopping the ball,
should the goalie stand at the point Y
on XY that lies on the perpendicular
bisector of GL or at the point on XY
that lies on the bisector of ∠GPL?
Explain your reasoning.

• How can you draw a diagram
to help?

• Would the goalie want to be the
same distance from G and L or

from PG and PL?

24. a. Constructions Draw ∠CDE. Construct the angle bisector of the angle.
b. Reasoning  Use the converse of the angle bisector theorem to justify your

construction.

25. a. Constructions Draw QR. Construct the perpendicular bisector of QR to
construct △PQR.

b. Reasoning  Use the perpendicular bisector theorem to justify that your

construction is an isosceles triangle.

26. Write Theorems 5-2 and 5-3 as a single biconditional statement.

27. Write Theorems 5-4 and 5-5 as a single biconditional statement.

Lesson 5-2  Perpendicular and Angle Bisectors 297

28. Error Analysis  To prove that △PQR is isosceles, a student began by stating P Q
that since Q is on the segment perpendicular to PR, Q is equidistant from the S
endpoints of PR. What is the error in the student’s reasoning?

Writing  Determine whether A must be on the bisector of jTXR. Explain. R

29. X 30. A 31. R
A hsm11gmXse_0502_t06118.ai
TR TR
8A 9 X T

32. Prove the Perpendicular hsm11gmse3_30.5 0P2ro_vt0e6th1e20C.oani verse of the

Proo f BGhiisvsemecnt1:o 1r<PgTmMhe>so#er_e0Am5B.0, 2<P_Mt0>6b1is1e9c.tasi AB Proo f PGeivrpeenn: dPicAu=larPBBisweichthtsoPmr MT1h1e#gormAeBmsae.t_M0.502_t06121.ai

Prove:  AP = BP Prove: P is on the perpendicular bisector

of AB.
P

P

AM B

AM B

34. Phrosmve1t1hgemAsneg_le0502_t06122.ai 35. Prove the Converse of the
Proof Bisector Theorem. Proo f AGhnivsgmelen1:B 1i SgsPemc#stoer_QT0Ph5>,e0So2Rr_et#m06.Q1R2>3, .ai
Given:  SQPS> #bisQePct>,sS∠RP#QRQ,R> SP = SR
Prove:  SP = SR
Prove:  QS> bisects ∠PQR.

P P

QS QS

RR

36. Coordinate Geometry  Use points Asu(6ch, 8t)h,aOt(/0,#0)<O, aAn>dahtBAs(1ma0n,1d01)mg. m#se<O_B0> a5t0B2._t06125.ai

ah. sWmri1te1egqmuasteio_n0s5o0f2li_nte0s6/1a2n4d.mai
b. Find the intersection C of lines / and m.
c. Show that CA = CB.
d. Explain why C is on the bisector of ∠AOB.

298 Chapter 5  Relationships Within Triangles

C Challenge 37. A, B, and C are three noncollinear points. Describe and sketch a E
line in plane ABC such that points A, B, and C are equidistant from
the line. Justify your response. A BM
ᐉ C
38. Reasoning  M is the intersection of the perpendicular bisectors of

two sides of △ABC. Line / is perpendicular to plane ABC at M. Explain
why a point E on / is equidistant from A, B, and C. (Hint: See page 48,
Exercise 33. Explain why △EAM ≅ △EBM ≅ △ECM.)

Standardized Test Prep hsm11gmse_0502_t06126.ai

SAT/ACT 39. For A(1, 3) and B(1, 9), which point lies on the perpendicular bisector of AB?

(3, 3) (1, 5) (6, 6) (3, 12)

40. What is the converse of the following conditional statement?
If a triangle is isosceles, then it has two congruent angles.
If a triangle is isosceles, then it has two congruent sides.
If a triangle has congruent sides, then it is equilateral.
If a triangle has two congruent angles, then it is isosceles.
If a triangle is not isosceles, then it does not have two congruent angles.

41. Which figure represents the statement BD bisects ∠ABC?
A A A A

B DB D B DB D
CC CC

Short 42. The line y = 7 is the perpendicular bisector of the segment with endpoints A(2, 10)
Response and B(2, k). What is the value of k? Explain your reasoning.

hsm11gmse_0502_t0h6s0m9511gmse_0502_t0h6s0m9611gmse_0502_t0h6s0m9711gmse_0502_t06099
Mixed Review

43. Find the value of x in the figure at the right. 2x ϩ 10 See Lesson 5-1.
3x Ϫ 7 See Lesson 1-5.
44. ∠1 and ∠2 are complementary and ∠1
and ∠3 are supplementary. If m∠2 = 30,
what is m∠3?

Get Ready!  To prepare for Lesson 5-3, do Exercises 45–47. See Lesson 3-8.

45. What is the slope of a line that is perpendicular htostmhe1l1ingemys=e_-035x0+24_?t06101

46. Line / is a horizontal line. What is the slope of a line perpendicular to /?

47. Describe the line x = 5.

Lesson 5-2  Perpendicular and Angle Bisectors 299

Concept Byte Paper Folding MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Bisectors
Use With Lesson 5-3 Prepares for MGA-CF.SA..931  2C.oGn-sCtr.u1c.3t  thCeoinnsstcrruibctedthaend
icnirsccurmibescdriabned circulems socfriabetrdiacnigrclele.s.o.f a triangle . . .
Activity
MP 5

In Activity 1, you will use paper folding to investigate the bisectors of the angles of
a triangle.

1

Step 1 Draw and cut out three different triangles: one acute, one right, and one obtuse. Folding an angle
bisector
Step 2 Use paper folding to make the angle bisectors of each angle of your acute
triangle. What do you notice about the angle bisectors?

Step 3 Repeat Step 2 with your right triangle and your obtuse triangle. Does your
discovery from Step 2 still hold true?

In Activity 2, you will use paper folding to investigate the perpendicular bisectors of hsm11gmse_0503a_t06486.ai
the sides of a triangle.

2

Step 1 Draw and cut out two different triangles: one acute and one right. Folding a perpendicular
bisector
Step 2 Use paper folding to make the perpendicular bisector of each side of your
acute triangle. What do you notice about the perpendicular bisectors?

Step 3 Repeat Step 1 with your right triangle. Does your discovery from Step 2 still
hold true?

Exercises hsm11gmse_0503a_t06493.ai

1. Make a Conjecture  Make a conjecture about the bisectors of the angles of a triangle.

2. Make a Conjecture  Make a conjecture about the perpendicular bisectors of the
sides of a triangle.

3. Extend  Draw and cut out an obtuse triangle. Fold the perpendicular bisectors.
a. How do the results for your obtuse triangle compare to the results for your

acute and right triangles from Activity 2?
b. Based on your answer to part (a), how would you revise your conjecture in Exercise 2?

4. Extend  For what type of triangle would the three perpendicular bisectors and the
three angle bisectors intersect at the same point?

300 Concept Byte  Paper Folding Bisectors

5-3 Bisectors in Triangles MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

MG-ACF.AS.39 1C2o.Gns-tCr.u1c.t3t hCeoinsctruibcetdthaendincsicrrciubmedscarnibded
circluems socfraibterdiacnigrclele.s.o.f a triangle . . .

MP 1, MP 3, MP 4, MP 7, MP 8

Objective To identify properties of perpendicular bisectors and angle bisectors

Can you conjecture Construct a circle and label its center C. Choose any three C
any other points on the circle and connect them to form a triangle.
properties that the Draw three lines from C such that each line is perpendicular
perpendicular lines to one side of the triangle. What conjecture can you make
might have? about the two segments into which each side of the triangle
is divided? Justify your reasoning.

MATHEMATICAL

PRACTICES

In the Solve It, the three lines you drew intersect at one point, the center of the circle.
When three or more lines intersect at one point, they are concurrent. The point at which
they intersect is the point of concurrency.

Lesson Essential Understanding  For any triangle, certain sets of lines are always
concurrent. Two of these sets of lines are the perpendicular bisectors of the triangle’s
Vocabulary three sides and the bisectors of the triangle’s three angles.
• concurrent
• point of Theorem 5-6  Concurrency of Perpendicular Bisectors Theorem

concurrency Theorem Diagram Symbols
• circumcenter of The perpendicular bisectors A Perpendicular bisectors
of the sides of a triangle PX, PY, and PZ are
a triangle are concurrent at a point X Y concurrent at P.
• circumscribed equidistant from the vertices. B P
Z PA = PB = PC
about
• incenter of a C

triangle
• inscribed in

The point of concurrency of the perpendicular bisectors A
P
of a triangle is called the circumcenter ohf stmhe1t1rigamngsle._0503_t06270.ai

Since the circumcenter is equidistant from the vertices, B C
you can use the circumcenter as the center of the circle
that contains each vertex of the triangle. You say the
circle is circumscribed about the triangle.

Lesson 5-3  Bisectors in Triangles 301

Proof Proof of Theorem 5-6 A
Given: Lines /, m, and n are the perpendicular bisectors m
of the sides of △ABC. P is the intersection of
ഞn

lines / and m. P

Prove: Line n contains point P, and PA = PB = PC. B C

Proof: A point on the perpendicular bisector of a segment

is equidistant from the endpoints of the segment. Point P is on /, which is
the perpendicular bisector of AB, so PA = PB. Using the same reasoning,

since P is on m, and m is the perpendicular bisector of BhCs,mP1B1=gPmCs.eT_h0u5s,03_t06271.ai

PA = PC by the Transitive Property. Since PA = PC, P is equidistant from

the endpoints of AC. Then, by the converse of the Perpendicular Bisector

Theorem, P is on line n, the perpendicular bisector of AC.

The circumcenter of a triangle can be inside, on, or outside a triangle.

Acute triangle Right triangle Obtuse triangle

Problem 1 Finding the Circumcenter of a Triangle
What arhestmhe1c1ogomrsdein_a0t5e0s3o_ftt0h6e2ch7is3rmc.au1im1cgemnster_o0f5t0h3e_ttr0i6a2n7gh5lse.mawi1it1hgvmersteic_e0s5P0(30_,t60)6,276.ai
O(0, 0), and S(4, 0)?

Find the intersection point of two of the triangle’s perpendicular bisectors. Here, it is
easiest to find the perpendicular bisectors of PO and OS.

Does the location Step 1 (0, 3) is the midpoint of y Step 2 (2, 0) is the midpoint
of the circumcenter PO. The line through (0, 3) 6P of OS. The line
make sense? that is perpendicular to through (2, 0) that is
Yes, △POS is a PO is y ϭ 3. 4 perpendicular to OS
right triangle, so its is x ϭ 2.

circumcenter should lie Step 3 Find the point where the Ϫ2 O Sx
two perpendicular bisectors 46
on its hypotenuse. intersect. x ϭ 2 and y ϭ 3
intersect at (2, 3).

The coordinates of the circumcenter of the triangle are (2, 3).
Got It? 1. What are the coordinates of the circumcenter of the triangle with vertices

hsm11gmAs(e2_,07)5,0B3(1_0t,076)2, a7n7d.aCi(10, 3)?

302 Chapter 5  Relationships Within Triangles

How do you find a Problem 2 Using a Circumcenter C
point equidistant A town planner wants to locate a new fire station equidistant Town Park
from three points? from the elementary, middle, and high schools. Where
As long as the three should he locate the station? A
points are noncollinear, B
they are vertices of Elementary School
a triangle. Find the
circumcenter of the
triangle.

E Middle School
M
High School
H

The three schools form the vertices of a triangle. The planner E

should locate the fire station at P, the point of concurrency
of the perpendicular bisectors of △EMH. This point is the
circumcenter of △EMH and is equidistant from the three

schools at E, M, and H.

Got It? 2. In Problem 2, the town planner wants to place a 2HnSdMp1Ha1sGsM0S1E-2_10-50093_a055P28 M
bench equidistant from the three trees in the parDku. rke
Where should he place the bench?

Theorem 5-7  Concurrency of Angle BisectohrssmT1h1eogrmemse_0503_t13166

Theorem Diagram Y Symbols
The bisectors of the B P Angle bisectors
angles of a triangle are AP, BP, and CP are
concurrent at a point X concurrent at P.
equidistant from the
sides of the triangle. PX = PY = PZ

AZ C

You will prove Theorem 5-7 in Exercise 24.

The point of concurrency of the angle bisectors of a triangle is called X B
the incenter of the triangle. Fhosrman1y1tgrmianseg_le0, 5th0e3_intc0e6n2t7e8r.iasi Y
always inside the triangle. In the diagram, points X, Y, and Z are
P
equidistant from P, the incenter of △ABC. P is the center of
Z
the circle that is inscribed in the triangle. A C

Lesson 5-3  Bisectors in Triangles 303

hsm 11gm se_0503_t06279.ai

Problem 3 Identifying and Using the Incenter of a Triangle A
Algebra  GE = 2x − 7 and GF = x + 4. What is GD?

What is the distance G is the incenter of △ABC because it is the point of D
from a point to a concurrency of the angle bisectors. By the Concurrency of E
line? Angle Bisectors Theorem, the distances from the incenter to
The distance from a point the three sides of the triangle are equal, so GE = GF = GD. G
to a line is the length Use this relationship to find x. C FB
of the perpendicular
segment that joins the 2x - 7 = x + 4 GE = GF hsm 11gm se_0503_t06280.ai
point to the line.
2x = x + 11 Add 7 to each side.

x = 11 Subtract x from each side.

Now find GF.

GF = x + 4

= 11 + 4 = 15 Substitute 11 for x.

Since GF = GD, GD = 15.

Got It? 3. a. QN = 5x + 36 and QM = 2x + 51. What is QO? K N
b. Reasoning  Is it possible for QP to equal 50? O P

Explain. Q L

J M

Lesson Check h s m 1 1 g m sMeA_T0HE5M0A3T_ICtA0L6 2 8 1 .a i
Do you UNDERSTAND? PRACTICES
Do you know HOW?
1. What are the coordinates of the circumcenter of the 3. Vocabulary  A triangle’s circumcenter is outside the

following triangle? triangle. What type of triangle is it?

y 4. Reasoning  You want to find the circumcenter of a
triangle. Why do you only need to find the
4 intersection of two of the triangle’s perpendicular
bisectors, instead of all three?
2
5. Error Analysis  Your TQ
x P
friend sees the triangle
O 246 C
at the right and R
concludes that CT = CP. S
What is the error in your
2. In the figure at the BU
friend’s reasoning?
right, TV = 3x - 12 C

and TU t=he5xvah-lsu2me4o1. f1xg?mse_0503_Tt06282.ai 6. Compare and Contrast  How are the circumcenter
What is

V and incenter of a triangle alike? How are they

A different? hsm11gmse_0503_t06284.ai

hsm11gmse_0503_t06283.ai
304 Chapter 5  Relationships Within Triangles

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Coordinate Geometry  Find the coordinates of the circumcenter of each See Problem 1.
triangle.

7. y 8. y
Ϫ2 O x 2

Ϫ2

Ϫ4 Ϫ2 O 2 x

4

Coordinate Geometry  Find the coordinates of the circumcenter of △ABC.

9. A(0, 0) 10. A(0, 0) 11. A( -4, 5) hsm111g2m. sAe(_-015,0-32_)t0 6286.1a3i . A(1, 4)

hBs(3m, 10)1gmse_0503_t0B6(42,805).ai B( -2, 5) B( -5, -2) B(1, 2)

C(3, 2) C(4, -3) C( -2, -2) C( -1, -7) C(6, 2)

14. City Planning  Copy the diagram of the beach. Show where town officials See Problem 2.

should place a recycling barrel so that it is equidistant from the lifeguard

chair, the snack bar, and the volleyball court. Explain.

Snack
bar

Volleyball Lifeguard
court chair

Name the point of concurrency of the angle bisectors. See Problem 3.

15. B 16.

C D X
A YZ
45Њ

Find the value of x. 18. RhsSm=141(Mgxm-s3e)_+0560a3n_dt0R6T28=8.5a(i2x - 6).
17. hsm11gmse_05043x _Ϫt016287.ai B 6x Ϫ 5 S

AC R
LT
N

hsm11gmse_0503_t06289.ai

Lesson 5-3  Bisectors in Triangles 305
hsm11gmse_0503_t06290.ai

B Apply 19. Think About a Plan  In the figure at the right, P is the incenter of isosceles △RST. S
What type of triangle is △RPT ? Explain. P
• What segments determine the incenter of a triangle?

• What do you know about the base angles of an isosceles triangle?

Constructions  Draw a triangle that fits the given description. Then construct R T

the inscribed circle and the circumscribed circle. Describe your method.

20. right triangle, △DEF hsm11gmse_0503_t06291.ai
21. obtuse triangle, △STU

# 22. Algebra  In the diagram at the right, G is the incenter of △DEF , E
m∠DEF = 60, and m∠EFD = 2 m∠EDF . What are m∠DGE, m∠DGF , G
and m∠EGF ?

23. Writing  Ivars found an old piece of paper inside an antique book. D F
It read,

From the spot I buried Olaf’s treasure, equal sets of paces did I measure; each of
three directions in a line, there to plant a seedling Norway pine. I could not return for
failing health; now the hounds of Haiti guard my wealth. —Karl

After searching Caribbean islands for five years, Ivars found an island with threhestamll11gmse_0503_t06535.ai
Norway pines. How might Ivars find where Karl buried Olaf’s treasure?

24. Use the diagram at the right to prove the Concurrrency of Angle B C
Proof Bisectors Theorem. Fᐍ

Given:  Rays /, m, and n are bisectors of the angles of △ABC. X is E Xn
the intersection of rays / and m, XD # AC, XE # AB, and
XF # BC. AD m

Prove:  Ray n contains point X, and XD = XE = XF .

25. Noise Control  You are trying to talk to a friend on the phone hsm11gmse_0503_t06292.ai
in a busy bus station. The buses are so loud that you can hardly
hear. Referring to the figure at the right, should you stand at P
or C to be as far as possible from all the buses? Explain.

Reasoning  Determine whether each statement is true or false. If C
the statement is false, give a counterexample. P

26. The incenter of a triangle is equidistant from all three vertices.

27. The incenter of a triangle always lies inside the triangle.

28. You can circumscribe a circle about any three points in a plane.

29. If point C is the circumcenter of △PQR and the circumcenter of △PQS, then R and
S must be the same point.

306 Chapter 5  Relationships Within Triangles

C Challenge 30. Reasoning  Explain why the circumcenter of a right triangle is on one of the
triangle’s sides.

Determine whether each statement is always, sometimes, or never true. Explain.
31. It is possible to find a point equidistant from three parallel lines in a plane.
32. The circles inscribed in and circumscribed about an isosceles triangle have the

same center.

Standardized Test Prep

SAT/ACT 33. Which of the following statements is false?

The bisectors of the angles of a triangle are concurrent.
The midsegments of a triangle are concurrent.
The perpendicular bisectors of the sides of a triangle are concurrent.
Four lines intersecting in one point are concurrent.

34. What type of triangle is △PUT ? T
right isosceles
obtuse scalene 42Њ

acute isosceles acute scalene P 45Њ

35. Which statement is logically equivalent to the following statement? U
If a triangle is right isosceles, then it has exactly two acute angles.

If a triangle is right isosceles, then it has one right angle.

If a triangle has exactly two acute angles, then it is right isosceles. hsm11gmse_0503_t06532.ai
K
If a triangle does not have exactly two acute angles, then it is not
right isosceles. V

If a triangle is not right isosceles, then it does not have a M

right angle.

Short 36. Refer to the figure at the right. Explain in two different ways why MV R
Response is the angle bisector of ∠KVR.

Mixed Review B7 C 3x ϩ 5 hsm11gmse_S0e5e03L_ets0s6o5n353-.a2.i
D
Use the figure at the right for Exercises 37 and 38. 2x Ϫ 1
37. Find the value of x.
38. Find the length of AD . A

G et Ready!  To prepare for Lesson 5-4, do Exercises 39 and 40.

Find the coordinates of the midpoint of AB with the given endpoints. See Lesson 1-7.
hsm11gmse_0503_t06534.ai
39. A(3, 0), B(3, 16) 40. A(6, 8), B(4, -1)

Lesson 5-3  Bisectors in Triangles 307

Concept Byte Special Segments MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
in Triangles
Use With Lesson 5-4 Prepares for GM-ACOFS.C.9.91 2P.Gro-vCeOt.h3e.o9r ePmrosvaebtohuetorems
atrbiaonugtletrsia.n.g. ltehse.m. .etdhiaenms eodfiantrsiaonfgaletrmiaenegtleatmaepetoiantt.a
technology pMoPint5.
MP 5

You already know about two sets of lines that are concurrent for any triangle. In the MATHEMATICAL
following activity, you will use geometry software to confirm what you know about the
concurrency of a triangle’s perpendicular bisectors and angle bisectors. Then you will PRACTICES
explore two more sets of special segments in triangles.
Altitude

Median
Use geometry software. hsm11gmse_0504a_t06495.ai

• Construct a triangle and the three perpendicular bisectors of its sides. Use your
result to confirm Theorem 5-6, the Concurrency of Perpendicular Bisectors Theorem.

• Construct a triangle and its three angle bisectors. Use your result to confirm
Theorem 5-7, the Concurrency of Angle Bisectors Theorem.

• An altitude of a triangle is the perpendicular segment from a vertex to the line
containing the opposite side. Construct a triangle. Through a vertex of the triangle
construct a segment that is perpendicular to the line containing the side opposite
that vertex. Next construct the altitudes from the other two vertices.

• A median of a triangle is the segment joining the midpoint of a side and the
opposite vertex. Construct a triangle. Construct the midpoint of one side. Draw the
median. Then construct the other two medians.

Exercises hsm11gmse_0504a_t06499.ai

1. What property do the lines containing altitudes and the medians seem to have?
Does the property still hold as you manipulate the triangles?

2. State your conjectures about the lines containing altitudes and about the medians
of a triangle.

3. Copy the table. Think about acute, right, and obtuse triangles. Use inside, on, or
outside to describe the location of each point of concurrency.

Perpendicular Angle Lines Containing Medians
Bisectors Bisectors the Altitudes ■
Acute Triangle ■ ■ ■
Right Triangle ■ ■ ■ ■
Obtuse Triangle ■ ■ ■
■

4. Extend  What observations, if any, can you make about these special segments for
isosceles triangles? For equilateral triangles?

hsm11gmse_0504a_t06501.ai
308 Concept Byte  Special Segments in Triangles

5-4 Medians and Altitudes MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

GM-ACFOS.C.9.120. GP-rCovOe.3th.1e0o rePmrosvaebtohuetotrreimansgalebsou. .t .
trhiea nmgeledsi;aunseotfhaeotrieamngslaebmoeuet ttraiat nagpleosintto. Asollsvoe
pGr-oSbRleTm.Bs.5. . . the medians of a triangle meet at a
pMoPint1.,AMlsPo3M, MAFPS5.9,1M2P.G7-S, MRTP.28.5
MP 1, MP 3, MP 5, MP 7, MP 8

Objective To identify properties of medians and altitudes of a triangle

Draw a large acute scalene △ABC. On B
each side, mark the point that is  15 of the
distance from one of the side’s endpoints,

as shown in the diagram. Connect each of

You can use these points to the opposite vertex.
different colors
for the sets Repeat this process for 41beanfdor13.12?WChhaetckdo A C
of segments so you think the result will
you can see the
pattern more your answer. Were you correct?
easily.

MATHEMATICAL In the Solve It, the last set of segments you drew are the triangle’s Median
medians. A median of a triangle is a segment whose endpoints are
PRACTICES

a vertex and the midpoint of the opposite side.

Essential Understanding  A triangle’s three medians are
always concurrent.

Theorem 5-8  Concurrency of Medians Theoremhsm11gmse_0504_t05777.ai

Lesson The medians of a triangle are concurrent at a point that is D

Vocabulary two thirds the distance from each vertex to the midpoint H
• median of a G
of the opposite side.
triangle C
• centroid of a DC = 2 DJ EC = 2 EG FC = 2 FH FJ E
3 3 3
triangle
• altitude of a You will prove Theorem 5-8 in Lesson 6-9.

triangle In a triangle, the point of concurrency of the medians is the centroid of the triangle.
• orthocenter of a The point is also called the center of gravity of a triangle because it is thehspmo1i1ngtmwshee_0re50a4_t05778.ai
triangular shape will balance. For any triangle, the centroid is always inside the triangle.
triangle

Lesson 5-4  Medians and Altitudes 309

Problem 1 Finding the Length of a Median

How do you use the In the diagram at the right, XA = 8. What is the length of XB? Y

centroid? A is the centroid of △XYZ because it is the point of concurrency of C B
Write an equation the triangle’s medians. A Z
relating the length of
the whole median to the XA = 2 XB Concurrency of Medians Theorem X
length of the segment 3
from the vertex to the 12
centroid. 8 = 2 XB Substitute 8 for XA.
3
( ) ( )3
8= 3 2 XB Multiply each side by 3 . 00000 0
2 2 3 2 11111 1

12 = XB Simplify. hsm11g23m23se32_03250324_32t05780.ai

44444 4

55555 5

66666 6

Got It? 1. a. In the diagram for Problem 1, ZA = 9. What is the length 77777 7
88888 8

of ZC? 99999 9

b. Reasoning  What is the ratio of ZA to AC? Explain.

An altitude of a triangle is the perpendicular segment from a vertex of the htrsima1n1gglemtsoe_0504_t05781.ai
the line containing the opposite side. An altitude of a triangle can be inside or outside
the triangle, or it can be a side of the triangle.

How do you Problem 2 Identifying Medians and Altitudes P
determine whether A For △PQS, is PR a median, an altitude, or neither? Explain. T
a segment is an
altitude or a median? PSQR,isthaesseigdme oenppt tohsaitteePxt.ePnRds#frQomR>,vseortPeRx P to the line containing R
is an altitude of △PQS.
Look at whether the
B For △PQS, is QT a median, an altitude, or neither? Explain. S Q
segment is perpendicular
QT is a segment that extends from vertex Q to the side opposite
to a side (altitude) and/or
Q. Since PT ≅ TS, T is the midpoint of PS. So QT is a median
bisects a side (median). of △PQS.

Got It? 2. For △ABC, is each segment a median, an altitude, or hsm11gmse_0504_t05782.ai
neither? Explain.
AF
a. AD b. EG c. CF EG

C DB

Theorem 5-9  Concurrency of Altitudes Theorem
The lines that contain the altitudes of a triangle are concurrent. hsm11gmse_0504_t05786.ai

You will prove Theorem 5-9 in Lesson 6-9.

310 Chapter 5  Relationships Within Triangles

The lines that contain the altitudes of a triangle are concurrent at the orthocenter of
the triangle. The orthocenter of a triangle can be inside, on, or outside the triangle.

Acute triangle Right triangle Obtuse triangle

hsPmr1o1gbmlseem_05034_t0F5i7n83d.aini g thehsmO1r1tghmoscee_0n5t0e4r_t05784.ai
△ABC has vertices A(1, 3), B(2, 7), and C(6, 3). What are the chosmor1d1ginmastee_s0o50f4th_te05785.ai
orthocenter of △ABC?

The coordinates of the three The intersection Write the equations of the lines that
vertices point of the contain two of the altitudes. Then solve
triangle’s altitudes the system of equations.

Which two altitudes Step 1 Find the equation of the line containing the altitude 8y B
should you choose? to AC. Since AC is horizontal, the line containing the
It does not matter, but altitude to AC is vertical. The line passes through the 6
vertex B(2, 7). The equation of the line is x = 2.
the altitude to AC is 4
a vertical line, so its Step 2 Find the equation of the line containing the --alt72it=ud-e1t.o
equation will be easy to BC. The slope of the line containing BC 2A C
find. x

is 3 O 246
6
Since the product of the slopes of two perpendicular

lines is -1, the line containing the altitude to BC has

slope 1.

The line passes through the vertex A(1, 3). The equation
of the line is y - 3 = 1(x - 1), which simplifies to y = x + 2.

Step 3 Find the orthocenter by solving this system of equations:  hxysm==1x21g+m2se_0504_t05787.ai
y = 2 + 2 Substitute 2 for x in the second equation.

y = 4 Simplify.

The coordinates of the orthocenter are (2, 4).

Got It? 3. △DEF has vertices D(1, 2), E(1, 6), and F(4, 2). What are the coordinates of
the orthocenter of △DEF ?

Lesson 5-4  Medians and Altitudes 311

Concept Summary  Special Segments and Lines in Triangles

Perpendicular Angle Bisectors Medians Altitudes
Bisectors Incenter Centroid Orthocenter

Circumcenter

hsm11gmse_0504_t05788.ahism11gmse_0504_t05790.aihsm11gmse_0504_t05789.ai

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Use △ABC for Exercises 1–4.
1. Is AP a median or an altitude? B 5. Error Analysis  Your classmate B
2. If AP = 18, what is KP?
3. If BK = 15, what is KQ? says she drew HJ as an altitude of H
4. Which two segments are altitudes? △ABC. What error did she make?
AK
Q P 6. Reasoning  Does it matter which A JC
two altitudes you use to locate the
orthocenter of a triangle? Explain.

C 7. Reasoning  The orthocenter of △ABC lies at vertex A.
What can you conclude about BA andhsAmC11?gEmxspel_a0i5n0.4_t05793.ai

hsm11gmse_0504_t05792.ai MATHEMATICAL

Practice and Problem-Solving Exercises PRACTICES

A Practice In △TUV, Y is the centroid. U W See Problem 1.
XY V
8. If YW = 9, find TY and TW. TZ See Problem 2.
B
9. If YU = 9, find ZY and ZU.

10. If VX = 9, find VY and YX.

For △ABC, is the red segment a median, an altitude, or neither? Explain.

11. A 12.h sm11gmse_0504C_t06247.ai 13.
A

B CB AC

312 Chapter 5  Relatiohnssmh1ip1sgmWsiet_h0in50T4r_iat0n6g2l4e9s.ai hsm11gmse_0504_t06253.ai hsm11gmse_0504_t06255.ai

Coordinate Geometry  Find the coordinates of the orthocenter of △ABC. See Problem 3.

14. A(0, 0) 15. A(2, 6) 16. A(0, -2)
B(4, 0) B(8, 6) B(4, -2)
C(4, 2) C(6, 2) C( -2, -8)

B Apply Name the centroid.

17. B 18. R

D G E LM
A H N
F C
PQ
Name the orthocenter of △XYZ.
20. Y
19. Y hsm11gmse_0504_t06237.ai
hsm11gmJse_0504_t06236.ai VU
XWZ
K Z
X

21. Think About a Plan  In the diagram at the right, QS and PT are P
altitudes and m∠R = 55. What is m∠POQ? S

••h smWW1hh1aagttmddsooe_ey0so5iu0t4mk_nte0oa6wn23afo8b.raoaiustethgme seunmt toofbteheananagltlietumdeea?shusrme1s1ignmse_050R4_t06240O.ai


a triangle? T Q

• How do you sketch overlapping triangles separately?

Constructions  Draw a triangle that fits the given description. Then construct
the centroid and the orthocenter.

22. acute scalene triangle, △LMN 23. obtuse isoscelehssmtr1ia1ngmglsee,_△05R0S4T_t06241.ai

In Exercises 24–27, name each segment. B
24. a median in △ABC
25. an altitude in △ABC FOE
26. a median in △BDC
27. an altitude in △AOC A P C
D

28. Reasoning  A centroid separates a median into two segments. What is the ratio of
the length of the shorter segment to the lengthhsomf 1th1egmlosne_g0e5r0s4e_gtm06e2n4t2?.ai

Lesson 5-4  Medians and Altitudes 313

Paper Folding  The figures below show how to construct altitudes and medians
by paper folding. Refer to them for Exercises 29 and 30.

Folding an Altitude Folding a Median

B QQ

AC P MR P MR

Fold the triangle so that a side AC Fold one vertex R to another Unfold the triangle. Then fold it so
overlaps itself and the fold contains vertex P. This locates the that the fold contains the midpoint M
the opposite vertex B. midpoint M of a side. and the opposite vertex Q.

29. Cut out a large triangle. Fold the paper carefully to construct the three medians of
the triangle and demonstrate the Concurrency of Medians Theorem. Use a
ruler to measure the length of each median and the distance of each vertex
from the centroid.

30. Cut out a large acute triangle. Fold the paper carefully to construct the three
altitudes of the triangle and demonstrate the Concurrency of Altitudes Theorem.

31. In the figure at the right, C is the centroid of △DEF . D
If GF = 12x2 + 6y, which expression represents CF?
G
6x2 + 3y 8x2 + 4y C

4x2 + 2y 8x2 + 3y H E

32. Reasoning  What type of triangle has its orthocenter on the exterior of F
the triangle? Draw a sketch to support your answer.

33. Writing  Explain why the median to the base of an isosceles triangle is
also an altitude.

34. Coordinate Geometry  △ABC has vertices A(0, 0), B(2, 6), and 6hsmy 11Bg(m2,s6e_) 0504_t06259.ai
C(8, 0). Complete the following steps to verify the Concurrency of

Medians Theorem for △ABC. 4L P M
2
ba.. FFiinndd ethqeucaotioorndsinoaf t<AesMo,>f <BmPN,itd,h> paeoniindntt<eCsrLLs>e., cMti,oannodfN<A.M> <BN>.
c. Find the coordinates of and x

d. TShhiosw pothinatt pisotihnet Pceinstoronid<C.L>. A 2 N 6 C(8, 0)

e. Use the Distance Formula to show that point P is two thirds of the distance from

each vertex to the midpoint of the opposite side.

C Challenge 35. Constructions  A, B, and O are three noncollinear points. Construct phosimnt1C1gsmusceh_0504_t06256.ai
that O is the orthocenter of △ABC. Describe your method.

36. Reasoning  In an isosceles triangle, show that the circumcenter, incenter, centroid,
and orthocenter can be four different points, but all four must be collinear.

314 Chapter 5  Relationships Within Triangles