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Find these vocabulary words in Lesson 4-1 and the Multilingual Glossary.
Classifying Triangles by Angle Measures
A. �PQS
A right angle has a measure of .
Since �QPS is a angle, �PQS is a triangle.
B. �PRQ
First, find m�QRP. Since �QRP and �SRP form a pair, the angles are
. To find m�QRP, subtract 54� from .
m�QRP � � 54� �
What kind of angle is �QRP ? . So, �PRQ is an triangle.
C. �PRS
First, find m�RPS.
Since �RPS and �RPQ form a angle, the angles are
.
To find m�RPS, subtract 23� from .
m�RPS � � 23� �
What kind of angle is �RPS? .
What kind of angles are �SRP and �PSR ?
So, �PRS is an triangle.
Classifying Triangles by Side Lengths
A. �JKL
How many sides are congruent in �JKL?
What kind of triangle is �JKL?
B. �KML
Find KL. How many sides are congruent, or have the same measure, in
�KML? So, what kind of triangle is �KML?
4AReady to Go On? Skills Intervention4-1 Classifying Triangles
Vocabulary
acute triangle equiangular triangle right triangle obtuse triangle
equilateral triangle isosceles triangle scalene triangle
Q
P
R
S59°
54°23°
31°
K
JM
L7 4
98
Copyright © by Holt, Rinehart and Winston. 41 Holt GeometryAll rights reserved.
Name Date Class
SECTION
Complementary angles have a sum of 90�.
A roofer is making repairs on the roof of a house. In order to be safe, he sets his ladder so that it makes a 15� angle with the house. What angle will his ladder make with the ground?
Understand the Problem
1. What angle does the ladder make with the house?
2. What angle does the house form with the ground?
3. What kind of triangle is �ABC ?
Mark the figure with the information given in the problem.
Make a Plan
4. The acute angles of a right triangle are complementary, so the sum of the
measures of the acute angles equals .
5. Complete: m� � m�C � 90�
6. Write an equation by substituting the known angle measures. � m� � 90�
Solve
7. Solve the equation you wrote in Exercise 6:
� m�C � 90�
� � Subtract 15 from both sides to isolate the variable.
m�C �
8. What angle does the roofer’s ladder make with the ground?
Look BackYou can check your work in two ways.
9. What is the sum of the angles in a triangle?
10. From your answer in Exercise 9, you know that m�A � m� � m� � .
11. Substitute the angle measures and check your work. 90� � 15� � �
Does your answer check?
12. To check using a second method, substitute your solution from Exercise 8
into the equation you wrote in Exercise 6: 15� � � 90�
Does your answer check?
4AReady to Go On? Problem Intervention4-2 Angle Relationships in Triangles
B
C Aground
ladder house
Name Date Class
Copyright © by Holt, Rinehart and Winston. 42 Holt GeometryAll rights reserved.
SECTION
Find these vocabulary words in Lesson 4-2 and the Multilingual Glossary.
Finding Angle Measures in TrianglesFind m�C.
By the Triangle Sum Theorem, the sum of the angle
measures in a triangle is .
In this triangle, m�A � m� � m� � 180�.
108� � � m�C � 180� Substitute known measures.
� m�C � 180� Add.
� � Subtract to isolate the variable.
m�C � Solve.
Finding Angle Measures in Right TrianglesOne of the acute angles in a right triangle measures 37.9�. What is the measure of the other acute angle?
Let the acute angles be �T and �U, with m�T � 37.9�.Since the measures of the acute angles in a right triangle are complementary,
m�T � m�U � . Substitute 37.9� for m�T and solve for m�U.
� m�U � �
� � Subtract to isolate the variable.
m�U � � Solve.
Applying the Exterior Angles TheoremFind m�Q.
Using the Exterior Angles Theorem, m� � m� � m�PRS. Substitute the given angle measures into the equation and solve for x.
� (5x � 3) �
� 5x � Add.
5x � 8x � Subtract 47 from both sides.
� �42 Subtract 8x from both sides.
x � Divide both sides by �3.
Substitute the value of x into (5x � 3) to find m�Q: (5x � 3) � (5)( ) � 3 �
4AReady To Go On? Skills Intervention4-2 Angle Relationships in Triangles
Vocabulary
Auxiliary line corollary interior exterior
interior angle exterior angle remote interior angle
108° 24°24°AA BB
CC
U
T
(8(8xx + 5)°+ 5)°xx (5(5xx + 3)°+ 3)°xx
44°44°
RRSS QQ
PP
4
Copyright © by Holt, Rinehart and Winston. 43 Holt GeometryAll rights reserved.
Name Date Class
SECTION
Ready To Go On? Skills Intervention4-3 Congruent Triangles
Find these vocabulary words in Lesson 4-3 and the Multilingual Glossary.
Naming Congruent Corresponding PartsGiven �BCD � �PQR. Identify the congruent corresponding parts to �B and
_ BD .
In a congruence statement, vertices are written in corresponding .
�B corresponds with � , so �B � � .
_
BD corresponds with , so _
BD � .
Using Corresponding Parts of Congruent TrianglesGiven �DEF � �WXY.
A. Find the value of m.
�D corresponds with � , so �D � � .
Since �D � � , m�D � m� .
Substitute values for the angle measures D and W. Solve to find the value of m.
87 �
�2 �2 Subtract 2 from both sides.
85 �
85 ___ 5 � _____ 5 Divide both sides by 5.
� m Solve for m.
B. Find DE.
First find the value of x. _
XY corresponds with , so _
XY � and XY � . XY � EF Substitute values for XY and EF and solve for x.
3x � 7 �
�7 �7 Add 7 to both sides.
3x �
3x ___ 3 � _____ 3 Divide both sides by 3.
x � Solve for x.
Substitute the value of x into DE and simplify. DE � 2x � 9 � 2( )� 9 �
4A
Vocabulary
corresponding angles corresponding sides congruent polygons
DD
EE FF26
WW
YYXX
2x – 9 87°
(5m + 2)°
3x – 7
Name Date Class
Copyright © by Holt, Rinehart and Winston. 44 Holt GeometryAll rights reserved.
SECTION
Copyright © by Holt, Rinehart and Winston. 45 Holt GeometryAll rights reserved.
Name Date Class
4-1 Classifying TrianglesClassify each triangle by its angle measures.
1. �QPR
2. �SRQ
3. �TRQ
Classify each triangle by its side lengths.
4. �QNM
5. �MPQ
6. �NLM
4-2 Angle Relationships in TrianglesFind each angle measure.
7. m�GFC 8. m�BAC
(8x + 19)°(12x – 7)°
26°
G F
C
D
(15x + 14)°(9x – 28)°
A C
B
D
9. A high school baseball team is designing a pennant with the school logo. The pennant is an isosceles triangle and the measure of the vertex angle is 46�.
Find the measure of the base angles.
Ready to Go On? Quiz4A
SECTION
60° 60°
30°
30°
T
P SR
Q
8
8
= =8
L P Q N
M
46°
4-3 Congruent TrianglesGiven �MNO � �GHI. Identify the congruent corresponding parts.
10. _
MO � 11. _
GH �
12. �N � 13. �G �
Given �ABC � �LMN. Find each value.
14. LM
15. x
16. Given: ‹
__ › RS �
‹
__ › UT , ‹
___ › UR �
‹
__ › TS , RS � UT,
_ UR �
_ TS
Prove: �URT � �STR
Complete the proof.
Statements Reasons
1. ‹
__ › RS �
‹
__ › UT 1.
2. �SRT � �UTR 2.
3. ‹
___ › UR �
‹
__ › TS 3.
4. 4. Alt Int. � Thm.
5. �RUT � �RST 5. Third � Thm.
6. RS � UT 6.
7. 7. Def. � segments
8. _
UR � _
TS 8.
9. 9. Reflex. Prop of �
10. �URT � �STR 10.
Ready to Go On? Quiz continued
4ASECTION
33°
A
C
B (3x)°
L
N
M
5t + 4
7t – 1
29
U T
SR
Copyright © by Holt, Rinehart and Winston. 46 Holt GeometryAll rights reserved.
Name Date Class
Copyright © by Holt, Rinehart and Winston. 47 Holt GeometryAll rights reserved.
Name Date Class
Ready to Go On? Enrichment
Exploring Exterior AnglesFor Exercises 1–4, find the angle measures.
1. m�ABD
2. m�BDC
3. m�BCD
4. m�BCE
5. What is the sum of the measures of the exterior angles of the triangle?
For Exercises 6–9, find the angle measures.
6. m�1 7. m�2
8. m�3 9. m�4
10. What is the sum of the measures of the exterior angles of the triangle?
For Exercises 11–17, find the indicated values.
11. x �
12. m�QSR
13. m�QSU
14. m�QRS 15. m�SRT
16. m�SQR 17. m�PQR
18. What is the sum of the measures of the exterior angles of the triangle?
19. Make a conjecture about the sum of the measures of the exterior angles of a triangle.
4A
118°
47°
E C
BA
D
14
3
38°
2
(15x – 7)°
(7x – 3)°
(x2 – 13)°U S
QP
R
T
SECTION
Name Date Class
Copyright © by Holt, Rinehart and Winston. 48 Holt GeometryAll rights reserved.
Find these vocabulary words in Lesson 4-4 and the Multilingual Glossary.
Using SSS and SAS to Prove Triangles Congruent_ JK � _
ML and _
JK � _
ML . Use SAS to explain why �JKM � �LMK.
It is given that _
JK � _
ML . This means that segment JK
is to segment ML. Mark this information on the figure.
It is given that _
JK � _
ML . This means that segment JK
is to segment ML. Mark this information on the figure.
Since _
JK � _
ML , you know that � � �LMK because of the
Theorem.
By the Reflexive Property of Congruence, you know that _
MK � .
Therefore, � � � by .
Proving Triangles CongruentGiven:
_ AB �
_ BC , _
DB bisects �ABC. Prove: �ABD � �CBD
It is given that _
AB � _
BC and _
DB bisects �ABC. Mark this information on the figure.
Since _
DB bisects �ABC, you know that � � �
because of the definition of an .
Enter this information in Step 2 of the proof.
By the Reflexive Property of Congruence, you know that � .Enter this information in Step 3 of the proof.
Therefore, you know that �ABD � �CBD by . Enter this information in Step 4 of the proof.
Statements Reasons
1. _
AB � _
BC , _
DB bisects �ABC 1. Given
2. 2.
3. 3.
4. �ABD � �CBD 4.
Ready to Go On? Skills Intervention4-4 Triangle Congruence: SSS and SAS4B
SECTION
Vocabulary
triangle rigidity included angle
C
BD
A
LM
KJ
Copyright © by Holt, Rinehart and Winston. 49 Holt GeometryAll rights reserved.
Name Date Class
Engineers often use triangles in designing structures because of their rigidity.
The figure shows a radio tower supported by cables of equal length. M is the midpoint of LN. Use SSS to explain why �PML � �PMN.
Understand the Problem
1. Why do you think a radio tower needs to be supported by cables?
2. Why do the cables form triangles with the tower and the ground?
3. The problem asks you to “Use SSS to explain why �PML � �PMN. When you explain something in Geometry, you must essentially write a paragraph proof. For every statement you make about the situation, you must also provide a
.
Make a PlanThe problem gives you information about the triangles that are formed by the tower, the cables, and the ground. Mark the figure with the given information as you answer each of the questions.
4. The sentence “The figure shows a tower supported by cables of equal length,”
tells you that � PN, and therefore, � .
5. The phrase “M is the midpoint of LN,” tells you that � .
6. The segment is congruent to itself.
SolveWrite a paragraph using the information you found in Exercises 4–6. Include justifications in your paragraph.
7. It is given that � , so � by the definition of
segments. By of a midpoint, � . By the
Property of Congruence, � . Therefore, �PML � �PMN by .
Look Back
8. To use the SSS Theorem to prove triangle congruence, 3 sides of one triangle
must be congruent to sides of a second triangle.
9. Have you proven that three sides of �PML are congruent to three sides of
�PMN? How?
Ready to Go On? Problem Solving Intervention4-4 Triangle Congruence: SSS and SAS4B
SECTION
L M N
P
Name Date Class
Copyright © by Holt, Rinehart and Winston. 50 Holt GeometryAll rights reserved.
Find this vocabulary word in Lesson 4-5 and the Multilingual Glossary.
Applying HL CongruenceDetermine if you can use the HL Congruence Theorem to prove the triangles congruent. Explain.
A. �QPR and �SRP According to the diagram, �QPR and �SRP are
triangles that share leg .
� by the Reflexive Property of Congruence. Is any information given to you about the hypotenuse of the right triangles?
This conclusion be proven by HL. You need to know that the
of the triangles are .
B. �CDE and �CBE
According to the diagram, �CDE and �CBE are
triangles that share hypotenuse .
� by the Reflexive Property of Congruence.
It is given that � , therefore � � � by HL.
Using AAS to Prove Triangles CongruentGiven: �J � �L,
_ JK � _
ML Prove: �JKM � �LMK
Mark the given information on the figure.Since it is given that
_ JK � _
ML , you know that � � � .
Because of the Property of Congruence, you know that � .
Therefore, you know that � � � because of AAS.Complete the flow-chart.
4BSECTION
Vocabulary
included side
Ready to Go On? Skills Intervention4-5 Triangle Congruence: ASA, AAS, and HL
RP
SQ
E
DB
C
– –
�J � �LGiven
Given
� �1. 2.AAS
2.
�4.
�3.
�
3.
LM
KJ
_JK ��
_ML
Copyright © by Holt, Rinehart and Winston. 51 Holt GeometryAll rights reserved.
Name Date Class
Ready to Go On? Skills Intervention4-6 Triangle Congruence: CPCTC4B
SECTION
Find this vocabulary word in Lesson 4-6 and the Multilingual Glossary.
Proving Corresponding Parts CongruentGiven: B is the midpoint of
_ AD ; _
AE � _
CD Prove:
_ BE �
_ BC
Mark the given information on the figure: B is the midpoint of _
AD and _
AE � _
CD .
Fill the given information into Step 1 and Step 3 of the flow-chart proof below.
Since B is the midpoint of _
AD , you know that � , because
of the definition of a .
Fill this information into Step 2 of the proof.
Since _
AE � _
CD , you know that � � �D and �E � �
because of the Angles Theorem.
Fill this information into Step 4 of your proof.
Therefore, �ABE � � by and � by CPCTC.
Fill this information into Steps 5 and 6 of your proof.
Complete the flow-chart:
Vocabulary
CPCTC
DE
CAB
� �4.
4.
�
� � �
Given
1.
2.
Given
3.
�2.
�6.
5.
5.
6.
� � �
Name Date Class
Copyright © by Holt, Rinehart and Winston. 52 Holt GeometryAll rights reserved.
Find this vocabulary word in Lesson 4-7 and the Multilingual Glossary.
Positioning a Figure in the Coordinate PlanePosition a right triangle with legs of 7 units and 2 units in the coordinate plane.
Use the origin as the vertex of the right angle.
Count spaces to the right to find a second vertex.
Count 2 units from the origin to find the third vertex.
Connect the vertices to form a right triangle. Label the vertices with their coordinates.
Assigning Coordinates to VerticesPosition square LMNO in the coordinate plane and give the coordinates of each vertex.
Use the origin as one vertex of the square. Label it L.
Draw another vertex on the x-axis, to the right of origin. Label this vertex M(a, 0).
Move the same distance up from the origin on the y-axis and label this vertex O(0, a).
Describe where to place vertex N.
What are the coordinates of this vertex?
Connect the vertices to form a square.
Writing a Coordinate ProofUse the square LMNO you drew above to prove that
_ LN � _
MO .
Complete and use the distance formula: d � ���
( � x 1 ) 2 � ( y 2 � ) 2
Substitute the coordinates of L and N into the distance formula to find LN. Simplify.
LN � ���
( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ���
(a � ) 2 � ( � 0 ) 2 �
Substitute the coordinates of M and O into the distance formula to find MO. Simplify.
MO � ���
( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ���
(0 � ) 2 � (a � ) 2 �
Does LN � MO ?
So, � _
MO because of the definition of congruent segments.
Ready to Go On? Skills Intervention4-7 Introduction to Coordinate Proof4B
SECTION
Vocabulary
coordinate proof
x
y
2
2 4 6
Find these vocabulary words in Lesson 4-8 and the Multilingual Glossary.
Finding the Measure of an AngleFind m�L.
Look at the diagram. What type of triangle is �JKL?
From the Isosceles Triangle Theorem, you know that
�L � � . Therefore, m� � m� .
m�L � m�K Substitute the given values and solve to find x.
7x � 4 �
7x � 16 � Add 12 to both sides.
16 � Subtract 7x from both sides.
x � Divide both sides by 2.
Substitute the value of x into m�L and simplify.
m�L � 7x � 4 � 7( ) � 4 � � 4 �
Using Coordinate ProofGiven: Isosceles �JKL has coordinates J(�2a, 0), K(2a, 0)
and L(0, 4b). M is the midpoint of _
JL , N is the midpoint of
_ KL , and O is the midpoint of
_ JK .
Prove: �MNO is isosceles.
Use the Midpoint Formula M � � x 1 + x 2 ______
2 ,
y 1 + y 2 ______
2 � to find the coordinates of M, N, and O.
Coordinates of M Coordinates of N Coordinates of O
M � � � 0 _________ 2 , � 4b ________
2 �
� (�a, )
N � � 0 � _________ 2 , � 0 ________ 2 � � (a, )
O � � � 2a __________ 2 , � 0 _______
2 �
� (0, )
Draw �MNO on the diagram above.
Substitute the coordinates into the Distance Formula and simplify to find OM and ON.
OM � ���
( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ���
( � 0 ) 2 � (2b � 0 ) 2 � ��
ON � ���
( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ���
(a � 0 ) 2 � ( � 0 ) 2 � ��
Does OM � ON ? Since OM � ON, by definition, � _
ON .
Therefore, �MNO is an triangle.
Copyright © by Holt, Rinehart and Winston. 53 Holt GeometryAll rights reserved.
Name Date Class
Ready to Go On? Skills Intervention4-8 Isosceles and Equilateral Triangles4B
SECTION
Vocabulary
legs of an isosceles triangle vertex angle base base angle
(7x + 4)x ° (9x – 12)°––
L K
J
––
J(–2JJ a, 0) K(2KK a, 0)
L(0, 4b)
Name Date Class
Copyright © by Holt, Rinehart and Winston. 54 Holt GeometryAll rights reserved.
4-4 Triangle Congruence SSS and SAS 1. The figure shows the logo used for a department store. Given
that _
KI bisects �HKJ and _
KH � _
KJ , use SAS to explain why �KIH � �KIJ.
2. Given: _
UV � _
TW , _
UV � _
TW Prove : �VUW � �TWU
4-5 Triangle Congruence ASA, AAS, and HLDetermine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know.
3. �ABD and �CDB 4. �NMO and �PMO
5. Use AAS to prove the triangles congruent. Given: K is the midpoint of
_ OM , _
ON � _
LM Prove: �LMK � �NOK
Ready to Go On? Quiz4B
SECTION
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4. WT
VU
H
I
J
K
M
PN
O
== ==
� �4.
4.
�
� � �
Given2.
Given
3.
.1 �2.
5.
�5.
CD
BA
M
N
L
O KK
4-6 Triangle Congruence CPCTC
6. Given _
TU � _
RS , _
TU � _
RS Prove:
_ QS �
_ QT
4-7 Introduction to Coordinate ProofPosition each figure in the coordinate plane.
7. a square with length 5 units 8. a right triangle with legs 5 units in length.
9. Assign coordinates to each vertex and write a coordinate proof Given: rectangle WXYZ Prove: WX � YZ
4-8 Isosceles and Equilateral TrianglesFind each angle measure.
10. m�Q (x + 4)°
(2x – 17)°––
Z Y
Q 11. m�E
48°==
––
C D
A
B
E
12. Given: Isosceles triangle LMN has coordinates L(0, 2b), M(2a, 0), and N(0, �2b). X is the midpoint of
_ LM and Y is the midpoint of
_ NM .
Prove: �XMY is isosceles.
Copyright © by Holt, Rinehart and Winston. 55 Holt GeometryAll rights reserved.
Name Date Class
Ready to Go On? Quiz continued
4BSECTION
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
T
S
U
R
Q
T
2–2 O–2
2
222222 2O2 2O–2 2O2 2O2 O
2
O 2–2 O–2
2
222222 2O2 2O–2 2O2 2O2 O
2
O
2–2 O–2
2
222222 2O2 2O–2 2O2 2O2 O
2
O
2–2 O–2
2
222222 2O2 2O–2 2O2 2O2 O
2
O
Name Date Class
Copyright © by Holt, Rinehart and Winston. 56 Holt GeometryAll rights reserved.
Trying Triangles
1. In the figure at the right, X is the midpoint of _
AB . Write a paragraph to explain whether or not �BXM � �AXN.
2. In the figure at the right, _
ML � _
NO , and m�MOL � (2x � 2)�. Find �NOM.
3. Figure ABCD has coordinates A(2, 5), B (5, 1), C (1, �2) and D (�2, 2). m�A � m�B � m�C � m�D. What type of figure is ABCD? Does AC � BD? Explain how you got your answers.
4. What kind of triangle is formed by the lines y � 9x � 32, x � y � �2, and x � 9y � �32? Explain your answer.
5. In the figure at right, _
CD � _
BA . Is this enough information to show that �BDC � �DBA? Explain your reasoning.
Ready to Go On? Enrichment4B
SECTION
A
M
N
B X6y –y 5y 2 – 5
y 2 – 8
4y +y 3
OL
NM(15x + 24y)°
(5x – 4y)°
–=
–
=
B A
DC
x
y
2 4O
2
–2–2–4
–4
4
AAA
B
C
D
22O
x
y
2 4O
2
–2–2–4
–4
4
OOO
Copyright © by Holt, Rinehart and Winston. 200 Holt GeometryAll rights reserved.
Copyright © by Holt, Rinehart and Winston. 38 Holt GeometryAll rights reserved.
3-5 Slopes of Lines Use the slope formula to determine the slope of each line.
1. ‹
__ › AD
� 1 __ 2
2. ‹
__ › AB
7 __ 2
3. ‹
__ › AC
2 __ 4
or 1 __ 2
4. ‹
__ › DB
� 10 ___ 4 or � 5 __ 2
Find the slope of the line through the given points.
5. R(4, 7) and S(�2, 0) 6. C(0, �4) and D(5, 9)
7 __ 6
13 ___ 5
7. H(3, 5) and I(�4, 2) 8. S(�6, 1) and T(3, �6)
3 __ 7
� 7 __ 9
Graph each pair of lines and use their slopes to determine if they are parallel, perpendicular, or neither.
9. ‹
___ › CD and
‹
__ › AB for A(�1, 0), B(1, 5), 10.
‹
__ › LM and
‹
___ › MN for L(�3, 2), M(�1, 5),
C(4, 5), and D(�2, 4) N(2, 3), and P(1, �5)
2 4–2–4 O
2
–2
4
–4
22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O
2222
4
–4
O
2 4–2–4 O
2
–2
4
–4
22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O
2222
4
–4
O
Neither Perpendicular
11. ‹
__ › PR and
‹
__ › PS for P(2, �1), Q(2, 1), 12.
‹
___ › GH and
‹
__ › FJ for F(�3, 2), G(�2, 5)
R(�3, 1), and S(�2, �2) H(2, 4), and J(2, 1)
4–2–4 O
2
–2
4
–4
2 4–2–4 O
2
–2
4
–4
Neither Parallel
Ready to Go On? Quiz3B
SECTION
x
y
2 4 6–2 O
2
–2
6
–4 B
D
C
A
––
4444
029-040_Ch3_RTGO_GEO_12738.indd 38 5/25/06 4:28:25 PM
3-6 Lines in the Coordinate PlaneWrite the equation of each line in the given form.
13. the line through (�3, �1) and (3, �3) in slope-intercept form y � � 1 __ 3 x � 2
14. the line through (6, �2) with slope � 3 __ 4 in point-slope form y � 2 � � 3 __
4 (x � 6)
15. the line with y-intercept �3 through the point (2, 5) in point-slope form y � 5 � 4(x � 2)
16. the line with x-intercept �4 and y-intercept 2 in slope-intercept form y � 1 __
2 x � 2
Graph each line.
17. y � 3x � 1 18. y � 1 � 3 __ 5
(x � 2) 19. y � �5
2 4–2–4
2
4
–4
2 4–2–4
2
4
–4
2 4–2–4 O–2
4
–4
2 4–2–4 O
2
–2
4
–4
22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O
2222
4
–4
O
Write the equation of each line.
20. 21. 22.
x
y
2–2–4 O
2
–2
4
–4
x
y
2–2–4 O
2
–2
4
x
y
2 4–2–4 O
2
–2
4
–4
x � 4 y � 3__
5 x � 3
y � �3
Determine whether the lines are parallel, intersect, or coincide.
23. y � � 1 __ 5
x � 2 24. 2x � 3y � 9 25. y � 5x � 3
x � 5y � 10 y � 2 __ 3
x � 1 y � 5x � 1
Coincide Intersect Parallel
Copyright © by Holt, Rinehart and Winston. 39 Holt GeometryAll rights reserved.
Ready to Go On? Quiz continued
3BSECTION
4444
029-040_Ch3_RTGO_GEO_12738.indd 39 10/13/05 9:45:37 AM
Copyright © by Holt, Rinehart and Winston. 40 Holt GeometryAll rights reserved.
Slopes and Lengths of Segments
Quadrilateral ABCD has vertices A(�5, 3), B(�1, 4), C(5, �3) and D(�4, �1).
1. Sketch and label the quadrilateral using the grid at the right.
2. Find the slopes of _
AC and _
BD . � 3 __
5 and 5 __
3
3. How are the segments related? They are perpendicular.
Quadrilateral PQRS has vertices P(2, 3), Q(2, �2), R(�2, �5), S(�2, 0). Use the information to answer the following questions:
4. Sketch and label the quadrilateral using the grid at the right.
Find the length of each segment.
5. Find PQ. 6. Find QR.
5 units 5 units
7. Find RS. 8. Find PS.
5 units 5 units
9. What can you conclude about the side lengths of the quadrilateral? They are congruent.
10. What is the slope of _
PR ? 11. What is the slope of _
QS ?
2 � 1 __ 2
12. What can you conclude about the diagonals of the quadrilateral? They are perpendicular.
13. Is the quadrilateral a square? Explain your answer. No; the sides do not meet at
right angles.
14. A triangle has vertices L(2, 8), M(5, 9), and N(4, 2). Write a paragraph proof to show that triangle LMN is a right triangle.Sample answer: Using the slope formula; the slope of
_ LM is 1 __
3 and the slope of
_ LN is �3. The product of the slopes is �1, so by the Perpendicular Lines Theorem,
the segments are perpendicular. Perpendicular lines form right angles, and by
definition, a right triangle is a triangle that has one right angle.
Ready to Go On? Enrichment3B
SECTION
2 4–2–4 O
2
–2
4
–4
2 4–2–4 O
2
–2
4
–4
22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O
2222
4
–4
O
44444444
444444444444
4
444 2222
22 44
44
4444
4444
44444444
–4–444
222222222222 222222222222
029-040_Ch3_RTGO_GEO_12738.indd 40 10/13/05 9:45:39 AM
Find these vocabulary words in Lesson 4-1 and the Multilingual Glossary.
Classifying Triangles by Angle Measures
A. �PQS
A right angle has a measure of 90� .
Since �QPS is a right angle, �PQS is a right triangle.
B. �PRQ
First, find m�QRP. Since �QRP and �SRP form a linear pair, the angles are
supplementary . To find m�QRP, subtract 54� from 180� .
m�QRP � 180� � 54� � 126�
What kind of angle is �QRP ? Obtuse . So, �PRQ is an obtuse triangle.
C. �PRS
First, find m�RPS.
Since �RPS and �RPQ form a right angle, the angles are
complementary .
To find m�RPS, subtract 23� from 90� .
m�RPS � 90� � 23� � 67�
What kind of angle is �RPS? Acute .
What kind of angles are �SRP and �PSR ? Acute
So, �PRS is an acute triangle.
Classifying Triangles by Side Lengths
A. �JKL
How many sides are congruent in �JKL? 2
What kind of triangle is �JKL? Isosceles
B. �KML
Find KL. 9 How many sides are congruent, or have the same measure, in
�KML? None So, what kind of triangle is �KML? Scalene
4AReady to Go On? Skills Intervention4-1 Classifying Triangles
Vocabulary
acute triangle equiangular triangle right triangle obtuse triangle
equilateral triangle isosceles triangle scalene triangle
Q
P
R
S59°
54°23°
31°
K
JM
L7 4
98
Copyright © by Holt, Rinehart and Winston. 41 Holt GeometryAll rights reserved.
SECTION
041-056_Ch004_RTGO_GEO_12738.indd 41 10/27/05 7:17:18 PM
Copyright © by Holt, Rinehart and Winston. 201 Holt GeometryAll rights reserved.
Complementary angles have a sum of 90�.
A roofer is making repairs on the roof of a house. In order to be safe, he sets his ladder so that it makes a 15� angle with the house. What angle will his ladder make with the ground?
Understand the Problem
1. What angle does the ladder make with the house? 15�
2. What angle does the house form with the ground? 90�
3. What kind of triangle is �ABC ? Right
Mark the figure with the information given in the problem.
Make a Plan
4. The acute angles of a right triangle are complementary, so the sum of the
measures of the acute angles equals 90� .
5. Complete: m� B � m�C � 90�
6. Write an equation by substituting the known angle measures. 15� � m� C � 90�
Solve
7. Solve the equation you wrote in Exercise 6:
15� � m�C � 90�
� 15� � 15� Subtract 15 from both sides to isolate the variable.
m�C � 75�
8. What angle does the roofer’s ladder make with the ground? 75�
Look BackYou can check your work in two ways.
9. What is the sum of the angles in a triangle? 180�
10. From your answer in Exercise 9, you know that m�A � m� B � m� C � 180� .
11. Substitute the angle measures and check your work. 90� � 15� � 75� � 180�
Does your answer check? Yes
12. To check using a second method, substitute your solution from Exercise 8
into the equation you wrote in Exercise 6: 15� � 75� � 90�
Does your answer check? Yes
4AReady to Go On? Problem Intervention4-2 Angle Relationships in Triangles
B
C Aground
ladder house
Copyright © by Holt, Rinehart and Winston. 42 Holt GeometryAll rights reserved.
SECTION
15°
041-056_Ch004_RTGO_GEO_12738.indd 42 10/27/05 7:17:22 PM
Find these vocabulary words in Lesson 4-2 and the Multilingual Glossary.
Finding Angle Measures in TrianglesFind m�C.
By the Triangle Sum Theorem, the sum of the angle
measures in a triangle is 180� .
In this triangle, m�A � m� B � m� C � 180�.
108� � 24� � m�C � 180� Substitute known measures.
132� � m�C � 180� Add.
� 132� � 132� Subtract to isolate the variable.
m�C � 48� Solve.
Finding Angle Measures in Right TrianglesOne of the acute angles in a right triangle measures 37.9�. What is the measure of the other acute angle?
Let the acute angles be �T and �U, with m�T � 37.9�.Since the measures of the acute angles in a right triangle are complementary,
m�T � m�U � 90� . Substitute 37.9� for m�T and solve for m�U.
37.9� � m�U � 90 �
� 37.9� � 37.9� Subtract to isolate the variable.
m�U � 52.1 � Solve.
Applying the Exterior Angles TheoremFind m�Q.
Using the Exterior Angles Theorem, m� P � m� Q � m�PRS. Substitute the given angle measures into the equation and solve for x.
44 � (5x � 3) � 8x � 5
47 � 5x � 8x � 5 Add.
5x � 8x � 42 Subtract 47 from both sides.
�3x � �42 Subtract 8x from both sides.
x � 14 Divide both sides by �3.
Substitute the value of x into (5x � 3) to find m�Q: (5x � 3) � (5)( 14 ) � 3 � 73�
4AReady To Go On? Skills Intervention4-2 Angle Relationships in Triangles
Vocabulary
Auxiliary line corollary interior exterior
interior angle exterior angle remote interior angle
108° 24°24°AA BB
CC
U
T
(8(8xx + 5)°+ 5)°xx (5(5xx + 3)°+ 3)°xx
44°44°
RRSS QQ
PP
4
Copyright © by Holt, Rinehart and Winston. 43 Holt GeometryAll rights reserved.
SECTION
041-056_Ch004_RTGO_GEO_12738.indd 43 10/27/05 7:17:25 PM
Ready To Go On? Skills Intervention4-3 Congruent Triangles
Find these vocabulary words in Lesson 4-3 and the Multilingual Glossary.
Naming Congruent Corresponding PartsGiven �BCD � �PQR. Identify the congruent corresponding parts to �B and
_ BD .
In a congruence statement, vertices are written in corresponding order .
�B corresponds with � P , so �B � � P .
_
BD corresponds with _
PR , so _
BD � _
PR .
Using Corresponding Parts of Congruent TrianglesGiven �DEF � �WXY.
A. Find the value of m.
�D corresponds with � W , so �D � � W .
Since �D � � W , m�D � m� W .
Substitute values for the angle measures D and W. Solve to find the value of m.
87 � 5m � 2
�2 �2 Subtract 2 from both sides.
85 � 5m
85 ___ 5 � 5m _____ 5 Divide both sides by 5.
17 � m Solve for m.
B. Find DE.
First find the value of x. _
XY corresponds with _
EF , so _
XY � _
EF and XY � EF . XY � EF Substitute values for XY and EF and solve for x.
3x � 7 � 26
�7 �7 Add 7 to both sides.
3x � 33
3x ___ 3 � 33 _____ 3 Divide both sides by 3.
x � 11 Solve for x.
Substitute the value of x into DE and simplify. DE � 2x � 9 � 2( 11 )� 9 � 13
4A
Vocabulary
corresponding angles corresponding sides congruent polygons
DD
EE FF26
WW
YYXX
2x – 9 87°
(5m + 2)°
3x – 7
Copyright © by Holt, Rinehart and Winston. 44 Holt GeometryAll rights reserved.
SECTION
041-056_Ch004_RTGO_GEO_12738.indd 44 10/27/05 7:17:28 PM
Copyright © by Holt, Rinehart and Winston. 45 Holt GeometryAll rights reserved.
4-1 Classifying TrianglesClassify each triangle by its angle measures.
1. �QPR Acute
2. �SRQ Obtuse
3. �TRQ Right
Classify each triangle by its side lengths.
4. �QNM Scalene
5. �MPQ Isosceles
6. �NLM Equilateral
4-2 Angle Relationships in TrianglesFind each angle measure.
7. m�GFC 149� 8. m�BAC 44�
(8x + 19)°(12x – 7)°
26°
G F
C
D
(15x + 14)°(9x – 28)°
A C
B
D
9. A high school baseball team is designing a pennant with the school logo. The pennant is an isosceles triangle and the measure of the vertex angle is 46�.
Find the measure of the base angles. 67�
Ready to Go On? Quiz4A
SECTION
60° 60°
30°
30°
T
P SR
Q
8
8
= =8
L P Q N
M
46°
041-056_Ch004_RTGO_GEO_12738.indd 45 10/27/05 7:17:29 PM
Copyright © by Holt, Rinehart and Winston. 202 Holt GeometryAll rights reserved.
4-3 Congruent TrianglesGiven �MNO � �GHI. Identify the congruent corresponding parts.
10. _
MO � _
GI 11. _
GH � _
MN
12. �N � �H 13. �G � �M
Given �ABC � �LMN. Find each value.
14. LM 34
15. x 11
16. Given: ‹
__ › RS �
‹
__ › UT , ‹
___ › UR �
‹
__ › TS , RS � UT,
_ UR �
_ TS
Prove: �URT � �STR
Complete the proof.
Statements Reasons
1. ‹
__ › RS �
‹
__ › UT 1. Given
2. �SRT � �UTR 2. Alt. Int. � Thm
3. ‹
___ › UR �
‹
__ › TS 3. Given
4. �URT � �RTS 4. Alt Int. � Thm.
5. �RUT � �RST 5. Third � Thm.
6. RS � UT 6. Given
7. _
RS � _
UT 7. Def. � segments
8. _
UR � _
TS 8. Given
9. _
RT � _
RT 9. Reflex. Prop of �
10. �URT � �STR 10. Def. of � �
Ready to Go On? Quiz continued
4ASECTION
33°
A
C
B (3x)°
L
N
M
5t + 4
7t – 1
29
U T
SR
s
Copyright © by Holt, Rinehart and Winston. 46 Holt GeometryAll rights reserved.
041-056_Ch004_RTGO_GEO_12738.indd 46 10/27/05 7:17:30 PM
Copyright © by Holt, Rinehart and Winston. 47 Holt GeometryAll rights reserved.
Ready to Go On? Enrichment
Exploring Exterior AnglesFor Exercises 1–4, find the angle measures.
1. m�ABD 133�
2. m�BDC 62�
3. m�BCD 71�
4. m�BCE 109�
5. What is the sum of the measures of the exterior angles of the triangle? 360�
For Exercises 6–9, find the angle measures.
6. m�1 7. m�2
52� 142�
8. m�3 9. m�4
90� 128�
10. What is the sum of the measures of the exterior angles of the triangle? 360�
For Exercises 11–17, find the indicated values.
11. x � 7
12. m�QSR 46�
13. m�QSU 134�
14. m�QRS 15. m�SRT
36� 144�
16. m�SQR 17. m�PQR
98� 82�
18. What is the sum of the measures of the exterior angles of the triangle? 360�
19. Make a conjecture about the sum of the measures of the exterior angles of a triangle.
Sample answer: The sum of the exterior angle measures of a triangle
is always 360�.
4A
118°
47°
E C
BA
D
14
3
38°
2
(15x – 7)°
(7x – 3)°
(x2 – 13)°U S
QP
R
T
SECTION
041-056_Ch004_RTGO_GEO_12738.indd 47 10/27/05 7:17:31 PM
Copyright © by Holt, Rinehart and Winston. 48 Holt GeometryAll rights reserved.
Find these vocabulary words in Lesson 4-4 and the Multilingual Glossary.
Using SSS and SAS to Prove Triangles Congruent_ JK � _
ML and _
JK � _
ML . Use SAS to explain why �JKM � �LMK.
It is given that _
JK � _
ML . This means that segment JK
is parallel to segment ML. Mark this information on the figure.
It is given that _
JK � _
ML . This means that segment JK
is congruent to segment ML. Mark this information on the figure.
Since _
JK � _
ML , you know that � JKM � �LMK because of the
Alternate Interior Angles Theorem.
By the Reflexive Property of Congruence, you know that _
MK � _
KM .
Therefore, � JKM � � LMK by SAS .
Proving Triangles CongruentGiven:
_ AB �
_ BC , _
DB bisects �ABC. Prove: �ABD � �CBD
It is given that _
AB � _
BC and _
DB bisects �ABC. Mark this information on the figure.
Since _
DB bisects �ABC, you know that � ABD � � CBD
because of the definition of an angle bisector .
Enter this information in Step 2 of the proof.
By the Reflexive Property of Congruence, you know that _
DB � _
DB .Enter this information in Step 3 of the proof.
Therefore, you know that �ABD � �CBD by SAS . Enter this information in Step 4 of the proof.
Statements Reasons
1. _
AB � _
BC , _
DB bisects �ABC 1. Given
2. �ABD � �CBD 2. Def. of � bisector3.
_ DB �
_ DB 3. Reflex. Prop. of �
4. �ABD � �CBD 4. SAS
Ready to Go On? Skills Intervention4-4 Triangle Congruence: SSS and SAS4B
SECTION
Vocabulary
triangle rigidity included angle
C
BD
A
LM
KJ
041-056_Ch004_RTGO_GEO_12738.indd 48 10/27/05 7:17:32 PM
Copyright © by Holt, Rinehart and Winston. 49 Holt GeometryAll rights reserved.
Engineers often use triangles in designing structures because of their rigidity.
The figure shows a radio tower supported by cables of equal length. M is the midpoint of LN. Use SSS to explain why �PML � �PMN.
Understand the Problem
1. Why do you think a radio tower needs to be supported by cables?
Sample answer: To protect it from the wind.
2. Why do the cables form triangles with the tower and the ground?
Sample answer: Because of triangle rigidity
3. The problem asks you to “Use SSS to explain why �PML � �PMN. When you explain something in Geometry, you must essentially write a paragraph proof. For every statement you make about the situation, you must also provide a
justification .
Make a PlanThe problem gives you information about the triangles that are formed by the tower, the cables, and the ground. Mark the figure with the given information as you answer each of the questions.
4. The sentence “The figure shows a tower supported by cables of equal length,”
tells you that PL � PN, and therefore, _
PL � _
PN .
5. The phrase “M is the midpoint of LN,” tells you that _
ML � _
MN .
6. The segment _
PM is congruent to itself.
SolveWrite a paragraph using the information you found in Exercises 4–6. Include justifications in your paragraph.
7. It is given that PL � PN , so _
PL � _
PN by the definition of congruent
segments. By definition of a midpoint, _
ML � _
MN . By the Reflexive
Property of Congruence, _
PM � _
PM . Therefore, �PML � �PMN by SSS .
Look Back
8. To use the SSS Theorem to prove triangle congruence, 3 sides of one triangle
must be congruent to 3 sides of a second triangle.
9. Have you proven that three sides of �PML are congruent to three sides of
�PMN? How? Yes, PM � PM , PL � PN , ML � MN
Ready to Go On? Problem Solving Intervention4-4 Triangle Congruence: SSS and SAS4B
SECTION
L M N
P
041-056_CH04_RTGO_GEO_12738.indd 49 5/25/06 4:29:22 PM
Copyright © by Holt, Rinehart and Winston. 203 Holt GeometryAll rights reserved.
Copyright © by Holt, Rinehart and Winston. 50 Holt GeometryAll rights reserved.
Find this vocabulary word in Lesson 4-5 and the Multilingual Glossary.
Applying HL CongruenceDetermine if you can use the HL Congruence Theorem to prove the triangles congruent. Explain.
A. QPR and SRP According to the diagram, QPR and SRP are righttriangles that share leg
_PR ._
PR_RP by the Reflexive Property of Congruence.
Is any information given to you about the hypotenuse of the right triangles? No
This conclusion cannot be proven by HL. You need to know that the hypotenuse of the triangles are congruent .
B. CDE and CBE
According to the diagram, CDE and CBE are righttriangles that share hypotenuse
_CE ._
CE_CE by the Reflexive Property of Congruence.
It is given that
_BE
_DE , therefore CDE CBE by HL.
Using AAS to Prove Triangles CongruentGiven: J L,
_JK
_ML
Prove: JKM LMK
Mark the given information on the figure.Since it is given that
_JK
_ML , you know that JKM LMK .
Because of the Reflexive Property of Congruence, you know that
_MK
_KM .
Therefore, you know that JKM LMK because of AAS.Complete the flow-chart.
4BSECTION
Vocabulary
included side
Ready to Go On? Skills Intervention4-5 Triangle Congruence: ASA, AAS, and HL
RP
SQ
E
DB
C
– –
J LGiven
Given
1. 2.AAS
2.
4.
3.
3.
LM
KJ
LMKJKM LMKJKM
_MK
_KM
Alt. Int. Thm
Refl. Prop. Of
_JK
_ML
041-056_CH04_RTGO_GEO_12738.indd 50 5/26/06 1:44:57 PM
Copyright © by Holt, Rinehart and Winston. 51 Holt GeometryAll rights reserved.
Ready to Go On? Skills Intervention4-6 Triangle Congruence: CPCTC4B
SECTION
Find this vocabulary word in Lesson 4-6 and the Multilingual Glossary.
Proving Corresponding Parts CongruentGiven: B is the midpoint of
_ AD ; _
AE � _
CD Prove:
_ BE �
_ BC
Mark the given information on the figure: B is the midpoint of _
AD and _
AE � _
CD .
Fill the given information into Step 1 and Step 3 of the flow-chart proof below.
Since B is the midpoint of _
AD , you know that _
AB � _
DB , because
of the definition of a midpoint .
Fill this information into Step 2 of the proof.
Since _
AE � _
CD , you know that � A � �D and �E � � C
because of the Alternate Interior Angles Theorem.
Fill this information into Step 4 of your proof.
Therefore, �ABE � � DBC by AAS and _
BE � _
BC by CPCTC.
Fill this information into Steps 5 and 6 of your proof.
Complete the flow-chart:
Vocabulary
CPCTC
DE
CAB
� �4.
4.
�
� � �
Given
1.
2.
Given
3.
�2.
�6.
5.
5.
6.
� � �
_
DB Def. of a Midpt.
_
AE � _
CD
_
AB B is the midpoint of AD
A D
E C
Alt. Int. � Thm
ABE DBC
AAS
_
BE
CPCTC
_
BC
041-056_Ch004_RTGO_GEO_12738.indd 51 10/27/05 7:17:40 PM
Copyright © by Holt, Rinehart and Winston. 52 Holt GeometryAll rights reserved.
Find this vocabulary word in Lesson 4-7 and the Multilingual Glossary.
Positioning a Figure in the Coordinate PlanePosition a right triangle with legs of 7 units and 2 units in the coordinate plane.
Use the origin as the vertex of the right angle.
Count 7 spaces to the right to find a second vertex.
Count 2 units up from the origin to find the third vertex.
Connect the vertices to form a right triangle. Label the vertices with their coordinates.
Assigning Coordinates to VerticesPosition square LMNO in the coordinate plane and give the coordinates of each vertex.
Use the origin as one vertex of the square. Label it L.
Draw another vertex on the x-axis, to the right of origin. Label this vertex M(a, 0).
Move the same distance up from the origin on the y-axis and label this vertex O(0, a).
Describe where to place vertex N. Sample answer: Vertex N will be a units up from vertex M and a units to the right of vertex O.
What are the coordinates of this vertex? The coordinates are (a, a).
Connect the vertices to form a square.
Writing a Coordinate ProofUse the square LMNO you drew above to prove that
_ LN � _
MO .
Complete and use the distance formula: d � ���
( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2
Substitute the coordinates of L and N into the distance formula to find LN. Simplify.
LN � ���
( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ���
(a � 0 ) 2 � ( a � 0 ) 2 � a ��
2
Substitute the coordinates of M and O into the distance formula to find MO. Simplify.
MO � ���
( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ���
(0 � a ) 2 � (a � 0 ) 2 � a ��
2
Does LN � MO ? Yes
So, _
LN � _
MO because of the definition of congruent segments.
Ready to Go On? Skills Intervention4-7 Introduction to Coordinate Proof4B
SECTION
Vocabulary
coordinate proof
x
y
2
2 4 6
041-056_Ch004_RTGO_GEO_12738.indd 52 10/27/05 7:17:43 PM
Find these vocabulary words in Lesson 4-8 and the Multilingual Glossary.
Finding the Measure of an AngleFind m�L.
Look at the diagram. What type of triangle is �JKL? Isosceles
From the Isosceles Triangle Theorem, you know that
�L � � K . Therefore, m� L � m� K .
m�L � m�K Substitute the given values and solve to find x.
7x � 4 � 9x � 127x � 16 � 9x Add 12 to both sides.
16 � 2x Subtract 7x from both sides.
x � 8 Divide both sides by 2.
Substitute the value of x into m�L and simplify.
m�L � 7x � 4 � 7( 8 ) � 4 � 56 � 4 � 60�
Using Coordinate ProofGiven: Isosceles �JKL has coordinates J(�2a, 0), K(2a, 0)
and L(0, 4b). M is the midpoint of _
JL , N is the midpoint of
_ KL , and O is the midpoint of
_ JK .
Prove: �MNO is isosceles.
Use the Midpoint Formula M � � x 1 + x 2 ______
2 ,
y 1 + y 2 ______
2 � to find the coordinates of M, N, and O.
Coordinates of M Coordinates of N Coordinates of O
M � � �2a � 0 _________ 2 , 0 � 4b ________
2 �
� (�a, 2b )
N � � 0 � 2a _________ 2 , 4b � 0 ________ 2 � � (a, 2b )
O � � �2a � 2a __________ 2 , 0 � 0 _______
2 �
� (0, 0 )
Draw �MNO on the diagram above.
Substitute the coordinates into the Distance Formula and simplify to find OM and ON.
OM � ���
( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ���
( �a � 0 ) 2 � (2b � 0 ) 2 � �� a 2 � 4 b 2
ON � ���
( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ���
(a � 0 ) 2 � ( 2b � 0 ) 2 � �� a 2 � 4 b 2
Does OM � ON ? Yes Since OM � ON, by definition, _
OM � _
ON .
Therefore, �MNO is an isosceles triangle.
Copyright © by Holt, Rinehart and Winston. 53 Holt GeometryAll rights reserved.
Ready to Go On? Skills Intervention4-8 Isosceles and Equilateral Triangles4B
SECTION
Vocabulary
legs of an isosceles triangle vertex angle base base angle
(7x + 4)x ° (9x – 12)°––
L K
J
––
J(–2JJ a, 0) K(2KK a, 0)
L(0, 4b)
O (0, 0)
N(NN a, 2b)M(–MM a, 2b)
041-056_Ch004_RTGO_GEO_12738.indd 53 10/27/05 7:17:47 PM
Copyright © by Holt, Rinehart and Winston. 204 Holt GeometryAll rights reserved.
Copyright © by Holt, Rinehart and Winston. 54 Holt GeometryAll rights reserved.
4-4 Triangle Congruence SSS and SAS 1. The figure shows the logo used for a department store. Given
that _
KI bisects �HKJ and _
KH � _
KJ , use SAS to explain why �KIH � �KIJ.
Sample answer: Since _
KI bisects �HKJ, �HKI � �JKI ;it is given that
_ KH �
_ KJ : _
KI � _
KI by the Reflexive Property of Congruence. So �KIH � �KIJ by SAS.
2. Given: _
UV � _
TW , _
UV � _
TW Prove : �VUW � �TWU
4-5 Triangle Congruence ASA, AAS, and HLDetermine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know.
3. �ABD and �CDB 4. �NMO and �PMO
Not enough information. Yes
Need to know that
AD � BC or _
AB � _
CD .
5. Use AAS to prove the triangles congruent. Given: K is the midpoint of
_ OM , _
ON � _
LM Prove: �LMK � �NOK
Ready to Go On? Quiz4B
SECTION
Statements Reasons
1. _
UV � _
TW , _
UV � _
TW 1. Given2. �VUW � �TWU 2. Alt. Int. � Thm3. _
UW � _
WU 3. Reflex. Prop. Of �4. �VUW � �TWU 4. SASWT
VU
H
I
J
K
M
PN
O
== ==
� �4.
4.
�
� � �
Given2.
Given
3.
.1 �2.
5.
�5.
CD
BA
M
N
L
O KK
K is mdpt. of _
OM _
OK _
MK
_
ON � _
LM AAS
Alt. Int. � Thm
�LMK �NOK
L NM O
Def. of mdpt.
041-056_CH04_RTGO_GEO_12738.indd 54 5/25/06 4:29:25 PM
4-6 Triangle Congruence CPCTC
6. Given _
TU � _
RS , _
TU � _
RS Prove:
_ QS �
_ QT
4-7 Introduction to Coordinate ProofPosition each figure in the coordinate plane. Check student’s graphs 7. a square with length 5 units 8. a right triangle with legs
5 units in length.
9. Assign coordinates to each vertex and write a coordinate proof Given: rectangle WXYZ Prove: WX � YZ
Check student’s graph and verify distance formula calculations.
4-8 Isosceles and Equilateral TrianglesFind each angle measure.
10. m�Q (x + 4)°
(2x – 17)°––
Z Y
Q 11. m�E
48°==
––
C D
A
B
E
46� 12�
12. Given: Isosceles triangle LMN has coordinates L(0, 2b), M(2a, 0), and N(0, �2b). X is the midpoint of
_ LM and Y is the midpoint of
_ NM .
Prove: �XMY is isosceles.
coordinate of X (a, b), coordinate of Y (a, �b),
MX � MY � ��
a 2 � b 2 Copyright © by Holt, Rinehart and Winston. 55 Holt GeometryAll rights reserved.
Ready to Go On? Quiz continued
4BSECTION
Statements Reasons
1. TU � RS; TU � RS 1. Given2. �T � �S 2. Alt. Int. � Thm3. �TQU � �SQR 3. Vert � Thm4. �TQU � �SQR 4. AAS5. QS � QT 5. CPCTC
T
S
U
R
Q
T
2–2 O–2
2
222222 2O2 2O–2 2O2 2O2 O
2
O 2–2 O–2
2
222222 2O2 2O–2 2O2 2O2 O
2
O
2–2 O–2
2
222222 2O2 2O–2 2O2 2O2 O
2
O
2–2 O–2
2
222222 2O2 2O–2 2O2 2O2 O
2
O
041-056_Ch004_RTGO_GEO_12738.indd 55 10/27/05 7:17:56 PM
Copyright © by Holt, Rinehart and Winston. 56 Holt GeometryAll rights reserved.
Trying Triangles
1. In the figure at the right, X is the midpoint of _
AB . Write a paragraph to explain whether or not �BXM � �AXN.
The triangles are not congruent. y � 6,
XA � XB � 31, XM � 27 but XN � 28.
2. In the figure at the right, _
ML � _
NO , and m�MOL � (2x � 2)�. Find �NOM.
38�
3. Figure ABCD has coordinates A(2, 5), B (5, 1), C (1, �2) and D (�2, 2). m�A � m�B � m�C � m�D. What type of figure is ABCD? Does AC � BD? Explain how you got your answers.
AB � BC � CD � AD � 5;
AC � BD � 5 ��
2 ; Since all angles are
90� and all sides have the same
lengths, then ABCD is a square.
4. What kind of triangle is formed by the lines y � 9x � 32, x � y � �2, and x � 9y � �32? Explain your answer.
This is an isosceles triangle. The
points of intersection of the lines are
(4, 4), (3, �5), and (�5, 3). The
lengths of the sides of the triangles
are ��
82 , ��
82 , and ��
128 � 8 ��
2 .
5. In the figure at right, _
CD � _
BA . Is this enough information to show that �BDC � �DBA? Explain your reasoning.
Sample answer: Yes. Given CD � BA, �CDB � �ABD by Alt. Int.
� Thm. �CDB � �CBD by Isoc. � Thm. �ABD � �CBD by subst.
�ABD � �ADB by Isoc. � Thm. �ADB � �CDB by subst. BD � BD
by Reflex. Prop. of �. �BDC � �DBA by ASA.
Ready to Go On? Enrichment4B
SECTION
A
M
N
B X6y –y 5y 2 – 5
y 2 – 8
4y +y 3
OL
NM(15x + 24y)°
(5x – 4y)°
–=
–
=
B A
DC
x
y
2 4O
2
–2–2–4
–4
4
AAAAAAA
BBB
CCC
DDD
2222OO
x
y
2 4O
2
–2–2–4
–4
4
OOOOOO
041-056_CH04_RTGO_GEO_12738.indd 56 5/26/06 2:37:44 PM
Find these vocabulary words in Lesson 5-1 and the Multilingual Glossary.
Applying the Perpendicular Bisector Theorem and Its ConverseFind each measure.
A. DF
Since CD � CE, and � � _
DE , ‹
___ › CF is the perpendicular
bisector of _
DE by the Converse of the
Perpendicular Bisector Theorem. Therefore, DF � FE because of the definition of a
segment bisector .
Substitute 3 for FE.
DF � 3
B. VU
TU � VU because of the Perpendicular Bisector Theorem. Substitute the given measures for TU and VU and solve for x.
12x � 3 � 14x � 5
12x � 12x � 3 � 5 � 14x � 5 � 5 � 12x
8 � 2 x
4 � x
Substitute the value of x to find VU. 14x � 5
� 14( 4 ) � 5
� 51
Applying the Angle Bisector TheoremFind QR.
QR � RS because of the Angle Bisector Theorem.Substitute 46 for RS.
QR � 46
Ready to Go On? Skills Intervention5-1 Perpendicular and Angle Bisectors
Vocabulary
equidistant focus
SECTION
Copyright © by Holt, Rinehart and Winston. 57 Holt GeometryAll rights reserved.
5A
C
D
E
F5.8
5.83
�
TU
V
W�
12x + 3
14x – 5
P
QR
S
4627°27°
�
057-075_Ch005_RTGO_GEO_12738.indd 57 11/1/05 7:22:32 PM
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