Download 4A Ready to Go On? Skills Intervention 4

Name
SECTION
4A
Date
Class
Ready to Go On? Skills Intervention
4-1 Classifying Triangles
Find these vocabulary words in Lesson 4-1 and the Multilingual Glossary.
Vocabulary
acute triangle
equiangular triangle
right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
Q
Classifying Triangles by Angle Measures
31°
A. 䉭PQS
R
.
A right angle has a measure of
angle, 䉭PQS is a
Since ⬔QPS is a
triangle.
54°
23°
B. 䉭PRQ
59°
P
First, find m⬔QRP. Since ⬔QRP and ⬔SRP form a
S
pair, the angles are
. To find m⬔QRP, subtract 54 from
.
54 m⬔QRP What kind of angle is ⬔QRP ?
. So, 䉭PRQ is an
triangle.
C. 䉭PRS
First, find m⬔RPS.
Since ⬔RPS and ⬔RPQ form a
angle, the angles are
.
.
To find m⬔RPS, subtract 23 from
23 m⬔RPS What kind of angle is ⬔RPS?
.
What kind of angles are ⬔SRP and ⬔PSR ?
So, 䉭PRS is an
triangle.
K
Classifying Triangles by Side Lengths
A. 䉭JKL
9
8
How many sides are congruent in 䉭JKL?
What kind of triangle is 䉭JKL?
J
B. ⬔KML
M 4
L
How many sides are congruent, or have the same measure, in
Find KL.
䉭KML?
7
So, what kind of triangle is 䉭KML?
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41
Holt Geometry
Name
SECTION
4A
Date
Class
Ready to Go On? Problem Intervention
4-2 Angle Relationships in Triangles
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
B
1. What angle does the ladder make with the house?
ladder
2. What angle does the house form with the ground?
3. What kind of triangle is 䉭ABC ?
C
house
ground
A
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
.
m⬔C 90
5. Complete: m⬔
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 Back
You 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?
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42
Holt Geometry
Name
SECTION
4A
Date
Class
Ready To Go On? Skills Intervention
4-2 Angle Relationships in Triangles
Find these vocabulary words in Lesson 4-2 and the Multilingual Glossary.
Vocabulary
Auxiliary line
corollary
interior
exterior
interior angle
exterior angle
remote interior angle
C
Finding Angle Measures in Triangles
Find mC.
By the Triangle Sum Theorem, the sum of the angle
measures in a triangle is
.
In this triangle, m⬔A m⬔
m⬔
108 180.
m⬔C 180
Substitute known measures.
m⬔C 180
Add.
108°
A
24°
B
Subtract to isolate the variable.
m⬔C Solve.
Finding Angle Measures in Right Triangles
One 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 .
m⬔U m⬔U Substitute 37.9 for m⬔T and solve for m⬔U.
T
Subtract to isolate the variable.
Solve.
U
Applying the Exterior Angles Theorem
Find mQ.
P
m⬔
m⬔PRS.
Using the Exterior Angles Theorem, m⬔
Substitute the given angle measures into the equation and solve for x.
44°
4
(5x 3) 5x Add.
(8xx + 5)°
(8
5x 8x Subtract 47 from both sides.
42
Subtract 8x from both sides.
x
R
Q
Divide both sides by 3.
Substitute the value of x into (5x 3) to find m⬔Q: (5x 3) (5)(
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S
(5xx + 3)°
(5
43
)3
Holt Geometry
Name
SECTION
4A
Date
Class
Ready To Go On? Skills Intervention
4-3 Congruent Triangles
Find these vocabulary words in Lesson 4-3 and the Multilingual Glossary.
Vocabulary
corresponding angles
corresponding sides
congruent polygons
Naming Congruent Corresponding Parts
Given BCD
_ PQR. Identify the congruent corresponding parts to
B and BD.
.
In a congruence statement, vertices are written in corresponding
, so ⬔B ⬔
⬔B corresponds with ⬔
_
.
_
, so BD BD corresponds with
.
Using Corresponding Parts of Congruent Triangles
Given DEF WXY.
A. Find the value of m.
, so ⬔D ⬔
⬔D corresponds with ⬔
Since ⬔D ⬔
, m⬔D m⬔
.
(5m + 2)° W
D
2x – 9 87°
.
E
26
3x – 7
F X
Y
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
m
Divide both sides by 5.
Solve for m.
B. Find DE.
_
_
, so XY and XY First find the value of x. XY corresponds with
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
x
Divide both sides by 3.
Solve for x.
Substitute the value of x into DE and simplify. DE 2x 9 2(
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44
) 9 Holt Geometry
Name
SECTION
Date
Class
Ready to Go On? Quiz
4A
Q
4-1 Classifying Triangles
Classify each triangle by its angle measures.
30°
1. 䉭QPR
T
2. 䉭SRQ
60°
3. 䉭TRQ
60°
P
30°
S
R
M
Classify each triangle by its side lengths.
4. 䉭QNM
8
5. 䉭MPQ
6. 䉭NLM
L
4-2 Angle Relationships in Triangles
Find each angle measure.
=
=
P
Q
8
N
8
8. m⬔BAC
7. m⬔GFC
C
B
26°
(9x – 28)°
A
(12x – 7)°
G
(15x + 14)°
C
D
(8x + 19)°
D
F
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.
46°
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45
Holt Geometry
Name
Date
SECTION
Class
Ready to Go On? Quiz continued
4A
4-3 Congruent Triangles
Given MNO GHI. Identify the congruent corresponding parts.
_
_
10. MO 11. GH 12. ⬔N 13. ⬔G Given ABC LMN. Find each value.
A
L
7t – 1
14. LM
33°
15. x
B
(3x)°
29
5t + 4
C
‹__›
‹__› ‹___›
‹__›
_
M
N
_
16. Given: RS UT, UR TS, RS UT, UR TS
R
S
Prove: URT STR
Complete the proof.
U
‹__›
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
‹___›
_
‹__›
T
_
8. UR TS
8.
9.
9. Reflex. Prop of 10. 䉭URT 䉭STR
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10.
46
Holt Geometry
Name
SECTION
Date
Class
Ready to Go On? Enrichment
4A
Exploring Exterior Angles
For Exercises 1–4, find the angle measures.
A
B
1. m⬔ABD
47°
2. m⬔BDC
3. m⬔BCD
E
4. m⬔BCE
118°
C
D
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
2
38°
8. m⬔3
9. m⬔4
3
1
4
10. What is the sum of the measures of the exterior angles of the triangle?
P
For Exercises 11–17, find the indicated values.
(15x – 7)°
Q
11. x 12. m⬔QSR
(7x – 3)°
U
13. m⬔QSU
14. m⬔QRS
15. m⬔SRT
16. m⬔SQR
17. m⬔PQR
S
R
(x 2 – 13)° T
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.
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47
Holt Geometry
Name
SECTION
4B
Date
Class
Ready to Go On? Skills Intervention
4-4 Triangle Congruence: SSS and SAS
Find these vocabulary words in Lesson 4-4 and the Multilingual Glossary.
Vocabulary
triangle rigidity
included angle
J
K
M
L
Using
_ SAS
_ to Prove Triangles Congruent
_ _SSS and
JK ML and JK ML. Use SAS to explain why JKM LMK.
_
_
It is given that JK ML. This means that segment JK
to segment ML. Mark this information on the figure.
is
_
_
It is given that JK ML. This means that segment JK
to segment ML. Mark this information on the figure.
is
_
_
Since JK ML, you know that ⬔
⬔LMK because of the
Theorem.
_
By the Reflexive Property of Congruence, you know that MK 䉭
Therefore, 䉭
by
.
.
A
Proving
_ _ Congruent
_Triangles
Given: AB BC, DB bisects ABC.
Prove: ABD CBD
_
_
D
_
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.
Therefore, you know that 䉭ABD 䉭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. 䉭ABD 䉭CBD
4.
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C
.
By the Reflexive Property of Congruence, you know that
Enter this information in Step 3 of the proof.
_
B
48
Holt Geometry
Name
SECTION
4B
Date
Class
Ready to Go On? Problem Solving Intervention
4-4 Triangle Congruence: SSS and SAS
Engineers often use triangles in designing structures because of their rigidity.
P
The figure shows a radio tower supported by cables of equal length.
M is the midpoint of LN. Use SSS to explain why 䉭PML ⬵ 䉭PMN.
Understand the Problem
1. Why do you think a radio tower needs to be supported by cables?
L
N
M
2. Why do the cables form triangles with the tower and the ground?
3. The problem asks you to “Use SSS to explain why 䉭PML ⬵ 䉭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 Plan
The 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,”
⫽ PN, and therefore,
tells you that
⬵
.
⬵
5. The phrase “M is the midpoint of LN,” tells you that
.
is congruent to itself.
6. The segment
Solve
Write a paragraph using the information you found in Exercises 4–6.
Include justifications in your paragraph.
7. It is given that
⫽
segments. By
Property of Congruence,
, so
⬵
by the definition of
⬵
of a midpoint,
⬵
. By the
. Therefore, 䉭PML ⬵ 䉭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 䉭PML are congruent to three sides of
䉭PMN? How?
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49
Holt Geometry
Name
Date
Class
SECTION
Ready to Go On? Skills Intervention
4B
4-5 Triangle Congruence: ASA, AAS, and HL
Find this vocabulary word in Lesson 4-5 and the
Multilingual Glossary.
Vocabulary
included side
Applying HL Congruence
Determine if you can use the HL Congruence Theorem to prove
the triangles congruent. Explain.
A. 䉭QPR and 䉭SRP
According to the diagram, 䉭QPR and 䉭SRP are
triangles that share leg
.
Q
S
P
R
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. 䉭CDE and 䉭CBE
C
According to the diagram, 䉭CDE and 䉭CBE are
.
triangles that share hypotenuse
B
D
–
–
by the Reflexive Property of Congruence.
It is given that
, therefore 䉭
䉭
J
Using AAS to Prove
Triangles Congruent
_ _
Given: ⬔J ⬵ ⬔L, JK 储 ML
Prove: 䉭JKM ⬵ 䉭LMK
Mark the given information
on the figure.
_ _
Since it is given that JK ML, you know that ⬔
Because of the
Therefore, you know that 䉭
Complete the flow-chart.
⬔
K
L
M
.
Property of Congruence, you know that
䉭
E
by HL.
.
because of AAS.
⬔J ⬔L
Given
_ _
1. JK 储 ML
Given
2. ⬔
⬔
4.
AAS
2.
3.
3.
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50
Holt Geometry
Name
SECTION
4B
Date
Class
Ready to Go On? Skills Intervention
4-6 Triangle Congruence: CPCTC
Find this vocabulary word in Lesson 4-6 and the
Multilingual Glossary.
Vocabulary
CPCTC
Proving Corresponding Parts
_
_ _Congruent
AD
;
AE
CD
Given: B
is
the
midpoint
of
_ _
Prove: BE BC
A
C
B
D
E
Mark
given_
information on the figure: B is the midpoint of
_ the_
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
of the definition of a
, because
.
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, 䉭ABE 䉭
by
and
by CPCTC.
Fill this information into Steps 5 and 6 of your proof.
Complete the flow-chart:
1.
2.
Given
3.
Given
5. 䉭
2.
4. ⬔
⬔
⬔
⬔
䉭
5.
6.
6.
4.
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51
Holt Geometry
Name
SECTION
4B
Date
Class
Ready to Go On? Skills Intervention
4-7 Introduction to Coordinate Proof
Find this vocabulary word in Lesson 4-7 and the
Multilingual Glossary.
Vocabulary
coordinate proof
Positioning a Figure in the Coordinate Plane
Position a right triangle with legs of 7 units and 2 units in
the coordinate plane.
y
2
x
Use the origin as the vertex of the right angle.
2
4
6 spaces to the right to find a second vertex.
Count
from the origin to find the third vertex.
Count 2 units
Connect the vertices to form a right triangle. Label the vertices with their
coordinates.
Assigning Coordinates to Vertices
Position 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 Proof
_ _
Use 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 2
2
(x 2 x 1) (y 2 y 1) (a )2 (
0) 2 Substitute the coordinates of M and O into the distance formula to find MO. Simplify.
MO 2
2
(x 2 x 1) (y 2 y 1) (0 ) 2 (a )2 Does LN MO ?
_
So,
MO because of the definition of congruent segments.
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52
Holt Geometry
Name
Date
SECTION
4B
Class
Ready to Go On? Skills Intervention
4-8 Isosceles and Equilateral Triangles
Find these vocabulary words in Lesson 4-8 and the Multilingual Glossary.
Vocabulary
legs of an isosceles triangle
vertex angle
base
base angle
Finding the Measure of an Angle
Find mL.
J
Look at the diagram. What type of triangle is 䉭JKL?
–
From the Isosceles Triangle Theorem, you know that
⬔L ⬔
. Therefore, m⬔
m⬔
.
m⬔L m⬔K
7x 4 7x 16 16 x
–
(9x – 12)°
K
(7xx + 4)°
L
Substitute the given values and solve to find x.
Add 12 to both sides.
Subtract 7x from both sides.
Divide both sides by 2.
Substitute the value of x into m⬔L and simplify.
m⬔L 7x 4 7(
)4
4
L(0, 4b)
Using Coordinate Proof
Given: Isosceles JKL has coordinates J(2a,
0), K(2a, 0)
_
the
and L(0, 4b)._
M is the midpoint of JL, N is_
midpoint of KL, and O is the midpoint of JK.
J a, 0)
J(–2
K a, 0)
K(2
Prove: MNO is isosceles.
+ y2
x 1 + x 2 y______
to find the coordinates of M, N, and O.
, 1
Use the Midpoint Formula M ______
2
2
Coordinates of M
Coordinates of N
Coordinates of O
0 , ________
4b
M _________
2
2
(a,
)
0
0
N _________, ________
2
2
(a,
)
0
2a , _______
O __________
2
2
(0,
)
Draw 䉭MNO on the diagram above.
Substitute the coordinates into the Distance Formula and simplify to find OM and ON.
2
2
2
2
0) (2b 0) OM (x 2 x 1) ( y 2 y 1) (
2
2
2
2
0) ON (x 2 x 1) ( y 2 y 1) (a 0) (
Does OM ON ?
Therefore, 䉭MNO is an
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All rights reserved.
Since OM ON, by definition,
_
ON .
triangle.
53
Holt Geometry
Name
SECTION
Date
Class
Ready to Go On? Quiz
4B
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
䉭KIH ⬵ 䉭KIJ.
K
I
H
_
_ _
_
2. Given: UV ⬵ TW , UV 储 TW
Prove : 䉭VUW ⬵ 䉭TWU
U
Statements
V
W
T
J
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
4-5 Triangle Congruence ASA, AAS, and HL
Determine if you can use the HL Congruence Theorem to prove the
triangles congruent. If not, tell what else you need to know.
3. 䉭ABD and 䉭CDB
O
B
=
=
A
4. 䉭NMO and 䉭PMO
N
C
D
M
L
5. Use AAS to prove the triangles
_congruent.
_ _
Given: K is the midpoint of OM , ON 储 LM
Prove: 䉭LMK ⬵ 䉭NOK
1.
2.
Given
3.
Given
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P
K
O
M
N
⬵
⬵
5.
2.
4. ⬔
⬔
⬵⬔
⬵⬔
54
5.
4.
Holt Geometry
Name
Date
SECTION
Class
Ready to Go On? Quiz continued
4B
4-6
Triangle Congruence CPCTC
_
_ _
_
, TU RS
6. Given TU
_
_ RS
T
Prove: QS QT
Statements
U
Q
R
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
S
4-7 Introduction to Coordinate Proof
Position each figure in the coordinate plane.
7. a square with length 5 units
8. a right triangle with legs
5 units in length.
2
–2
2O
–2
2
2
2
–2
2O
–2
2
9. Assign coordinates to each vertex and write a coordinate proof
Given: rectangle WXYZ
Prove: WX YZ
2
2
–2
2O
–2
2
2
4-8 Isosceles and Equilateral Triangles
Find each angle measure.
10. m⬔Q
E
11. m⬔E
Q (x + 4)°
–
–
A
–
Z
B
=
(2x – 17)°
Y
–
=
48°
C
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: 䉭XMY is isosceles.
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55
D
2
–2
2O
–2
2
2
Holt Geometry
Name
SECTION
Date
Class
Ready to Go On? Enrichment
4B
N
y2 – 8
Trying Triangles
_
1. In the figure at the right, X is the midpoint of AB .
Write a paragraph to explain whether or not
䉭BXM ⬵ 䉭AXN.
6yy – 5
B
X
A
y2 – 5
4yy + 3
M
_
(15x + 24y)°
M
_
N
(5x – 4y)°
2. In the figure at the right, ML ⬵ NO , and
m⬔MOL (2x 2). Find ⬔NOM.
O
L
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.
y
A
4
D
2
–4 –2 O
–2
–4
B
2
x
4
C
y
4. What kind of triangle is formed by the lines
y 9x 32, x y 2, and x 9y 32?
Explain your answer.
4
2
x
–4 –2 O
–2
2
4
–4
_
C
=
_
5. In the figure at right, CD 储 BA. Is this enough information to show
that 䉭BDC ⬵ 䉭DBA? Explain your reasoning.
D
–
B
Copyright © by Holt, Rinehart and Winston.
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56
–
=
A
Holt Geometry
SECTION
Ready to Go On? Quiz
SECTION
3B
Ready to Go On? Quiz continued
3B
3-5 Slopes of Lines
Use the slope formula to determine the slope of each line.
‹__›
‹__›
2. AB
2
C
4 2
10
5
⫺___ or ⫺__
4
2
‹__›
4. DB
x
–2 O
–2
2
4
3
14. the line through (6, ⫺2) with slope ⫺__ in point-slope form
4
⫺ 5 ⫽ 4(x ⫺ 2)
1x ⫹ 2
y ⫽ __
2
15. the line with y-intercept ⫺3 through the point (2, 5) in point-slope form y
6
B
–4
1
y ⫽ ⫺__x ⫺ 2
3
3(x ⫺ 6)
y ⫹ 2 ⫽ ⫺__
4
13. the line through (⫺3, ⫺1) and (3, ⫺3) in slope-intercept form
A
1
2 or __
__
‹__›
3. AC
6
D
2
7
__
2
1. AD
3-6 Lines in the Coordinate Plane
Write the equation of each line in the given form.
y
1
⫺__
16. the line with x-intercept ⫺4 and y-intercept 2 in slope-intercept form
Graph each line.
Find the slope of the line through the given points.
3 (x ⫹ 2)
18. y ⫺ 1 ⫽ __
5
17. y ⫽ 3x ⫺ 1
6. C(0, ⫺4) and D(5, 9)
5. R(4, 7) and S(⫺2, 0)
19. y ⫽ ⫺5
13
___
7
__
5
6
4
3
__
7
⫺__
9
7
4
4
2
8. S(⫺6, 1) and T(3, ⫺6)
7. H(3, 5) and I(⫺4, 2)
2
–4 –2
2
–4 –2 O
–2
4
–4
4
2
–4 –2
–4
2O
–2
2
4
2
4
–4
–4
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),
C(4, 5), and D(⫺2, 4)
Write the equation of each line.
‹___›
10. LM and MN for L(⫺3, 2), M(⫺1, 5),
N(2, 3), and P(1, ⫺5)
20.
21.
22.
y
2
–4 –2
–4
2O
–2
2
4
2
Neither
‹__›
‹___›
11. PR and PS for P(2, ⫺1), Q(2, 1),
R(⫺3, 1), and S(⫺2, ⫺2)
12. GH and FJ for F(⫺3, 2), G(⫺2, 5)
H(2, 4), and J(2, 1)
4
2
2
–4 –2 O
–2
–4 –2 O
–2
4
1x ⫹ 2
23. y ⫽ ⫺__
5
4
x
y ⫽ ⫺3
2
24. 2x ⫹ 3y ⫽ 9
4
25. y ⫽ 5x ⫺ 3
2x ⫺ 1
y ⫽ __
3
x ⫹ 5y ⫽ 10
Coincide
–4
y ⫽ 5x ⫹ 1
Intersect
Parallel
Parallel
38
Copyright © by Holt, Rinehart and Winston.
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Holt Geometry
Ready to Go On? Enrichment
39
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
SECTION
3B
4A
029-040_Ch3_RTGO_GEO_12738.indd 38
Holt Geometry
Ready to Go On? Skills Intervention
4-1 Classifying Triangles
029-040_Ch3_RTGO_GEO_12738.indd
5/25/06 394:28:25 PM
Quadrilateral ABCD has vertices A(⫺5, 3), B(⫺1, 4), C(5, ⫺3) and
D(⫺4, ⫺1).
_
2. Find the slopes of AC and BD .
3. How are the segments related?
Vocabulary
2
–4
4 –2
2O
$
2
–2
3
5
⫺__ and __
5
"
4
!
1. Sketch and label the quadrilateral using the grid at the right.
_
2
4
–4
3
acute triangle
equiangular triangle
right triangle
equilateral triangle
isosceles triangle
scalene triangle
obtuse triangle
#
Q
Classifying Triangles by Angle Measures
They are perpendicular.
31°
A. 䉭PQS
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
6. Find QR.
5 units
5 units
7. Find RS.
S
right
angle, the angles are
complementary .
_
_
11. What is the slope of QS ?
90⬚
67⬚
Acute .
.
To find m⬔RPS, subtract 23⬚ from
1
⫺__
2
m⬔RPS ⫽
90⬚
⫺ 23⬚ ⫽
What kind of angle is ⬔RPS?
12. What can you conclude about the diagonals of the quadrilateral? They are perpendicular.
13. Is the quadrilateral a square? Explain your answer.
59°
P
Since ⬔RPS and ⬔RPQ form a
What can you conclude about the side lengths of the quadrilateral? They are congruent.
2
54°
23°
First, find m⬔RPS.
5 units
10. What is the slope of PR ?
triangle.
C. 䉭PRS
8. Find PS.
5 units
right
angle, 䉭PQS is a
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, 䉭PRQ is an obtuse triangle.
2
5. Find PQ.
right
Since ⬔QPS is a
R
.
B. 䉭PRQ
2 4
1
–4
4
Find the length of each segment.
90⬚
A right angle has a measure of
0
2
3
–4 –2
–4
2O
–2
2
4. Sketch and label the quadrilateral using the grid at the right.
What kind of angles are ⬔SRP and ⬔PSR ?
No; the sides do not meet at
So, 䉭PRS is an
acute
Acute
triangle.
K
right angles.
Classifying Triangles by Side Lengths
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.
1 and the slope of
Sample answer: Using the slope formula; the slope of LM is __
A. 䉭JKL
LN is ⫺3. The product of the slopes is ⫺1, so by the Perpendicular Lines Theorem,
What kind of triangle is 䉭JKL?
Find KL.
definition, a right triangle is a triangle that has one right angle.
䉭KML?
40
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Holt Geometry
9
None
J
200
7
M 4
L
How many sides are congruent, or have the same measure, in
So, what kind of triangle is 䉭KML?
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
041-056_Ch004_RTGO_GEO_12738.indd
10/13/05 9:45:39
41
AM
8
Isosceles
B. ⬔KML
the segments are perpendicular. Perpendicular lines form right angles, and by
9
2
How many sides are congruent in 䉭JKL?
3
_
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
10/13/05 9:45:37 AM
Find these vocabulary words in Lesson 4-1 and the Multilingual Glossary.
Slopes and Lengths of Segments
029-040_Ch3_RTGO_GEO_12738.indd 40
2
Determine whether the lines are parallel, intersect, or coincide.
4
Neither
9.
y ⫽ _3_x ⫺ 3
5
x⫽4
‹__›
–4
SECTION
–4 –2 O
–2
x
2
–4
Perpendicular
‹__›
2
–4 –2 O
–2
2
–4
–4
–4
4
2
x
–4 –2 O
–2
4
y
4
2
2
2
–4 –2
–4
2O
–2
–
2
4
4
4
y
41
Scalene
Holt Geometry
Holt Geometry
10/27/05 7:17:18 PM
SECTION
4A
Ready to Go On? Problem Intervention
4A
4-2 Angle Relationships in Triangles
Find these vocabulary words in Lesson 4-2 and the Multilingual Glossary.
Complementary angles have a sum of 90.
Vocabulary
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
B
15°
15
90
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 ?
Ready To Go On? Skills Intervention
SECTION
4-2 Angle Relationships in Triangles
ladder
Right
C
corollary
interior
interior angle
exterior angle
remote interior angle
24
132
132
108 4. The acute angles of a right triangle are complementary, so the sum of the
90
5. Complete: m
.
15
6. Write an equation by substituting the known angle measures.
m
C
90
7. Solve the equation you wrote in Exercise 6:
180.
mC 180
Add.
132
48
108°
A
mT mU Look Back
You can check your work in two ways.
180
B
m
C
75
11. Substitute the angle measures and check your work. 90 15 Subtract to isolate the variable.
Solve.
.
Substitute 37.9 for mT and solve for mU.
T
Subtract to isolate the variable.
Solve.
U
44
Yes
8x 5
8x 5
5x 8x 42
3x 42
x 14
90
Yes
5x Add.
(8xx + 5)°
(8
S
Subtract 47 from both sides.
4A
Holt Geometry
Ready To Go On? Skills Intervention
corresponding sides
congruent polygons
Naming Congruent Corresponding Parts
Given BCD
_ PQR. Identify the congruent corresponding parts to
B and BD.
BD corresponds with
PR
_P
, so B _
, so BD PR
.
.
Using Corresponding Parts of Congruent Triangles
Given DEF WXY.
A. Find the value of m.
D corresponds with Since D W
W
, so D , mD m
W
Ready to Go On? Quiz
W
(5m + 2)° W
D
2x – 9 87°
E
26
1. QPR
Acute
2. SRQ
Obtuse
3. TRQ
Right
30°
60°
3x – 7
F X
4. QNM
Scalene
5. MPQ
Isosceles
6. NLM
Equilateral
8
5m 2
2
2
Subtract 2 from both sides.
85 5m
5m
_____
Divide both sides by 5.
17
5
m
7
149
8. mBAC
A
(12x – 7)°
_
_
G
_
3x x
44
.
(15x + 14)°
C
D
(8x + 19)°
D
F
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
Add 7 to both sides.
33
33
3x _____
___
3
EF
26
7
N
8
B
First find the value of x. XY corresponds with EF , so XY EF and XY XY EF
Substitute values for XY and EF and solve for x.
3x 7 Q
8
26°
Solve for m.
_
=
P
(9x – 28)°
B. Find DE.
=
C
87 5
S
M
Substitute values for the angle measures D and W. Solve to find the value of m.
85
___
30°
R
L
7. mGFC
Y
T
60°
P
4-2 Angle Relationships in Triangles
Find each angle measure.
.
.
10/27/05 7:17:25 PM
Q
Classify each triangle by its side lengths.
order .
In a congruence statement, vertices are written in corresponding
_
73
Holt Geometry
4-1 Classifying Triangles
Classify each triangle by its angle measures.
Vocabulary
_P
)3
041-056_Ch004_RTGO_GEO_12738.indd
10/27/05 7:17:22
43
PM
Find these vocabulary words in Lesson 4-3 and the Multilingual Glossary.
B corresponds with 14
43
4A
4-3 Congruent Triangles
041-056_Ch004_RTGO_GEO_12738.indd 42
corresponding angles
Q
Divide both sides by 3.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
SECTION
(5xx + 3)°
(5
R
Subtract 8x from both sides.
Substitute the value of x into (5x 3) to find mQ: (5x 3) (5)(
42
44°
4
(5x 3) 47
75
P
Using the Exterior Angles Theorem, m P m Q mPRS.
Substitute the given angle measures into the equation and solve for x.
180 .
180
12. To check using a second method, substitute your solution from Exercise 8
SECTION
B
Applying the Exterior Angles Theorem
Find mQ.
10. From your answer in Exercise 9, you know that mA m
into the equation you wrote in Exercise 6: 15 90
37.9 mU 90 37.9
37.9
mU 52.1 75
8. What angle does the roofer’s ladder make with the ground?
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
24°
Let the acute angles be T and U, with mT 37.9.
Since the measures of the acute angles in a right triangle are complementary,
Subtract 15 from both sides to isolate the variable.
Does your answer check?
C
Finding Angle Measures in Right Triangles
One of the acute angles in a right triangle measures 37.9. What is
the measure of the other acute angle?
Solve
Does your answer check?
m
Substitute known measures.
mC 9. What is the sum of the angles in a triangle?
.
mC 180
mC 90
15 mC 90
15
15
mC 75
180
B
measures in a triangle is
In this triangle, mA m
Make a Plan
B
C
By the Triangle Sum Theorem, the sum of the angle
A
Mark the figure with the information given in the problem.
measures of the acute angles equals
exterior
Finding Angle Measures in Triangles
Find mC.
house
ground
Auxiliary line
3
11
Solve for x.
Substitute the value of x into DE and simplify. DE 2x 9 2(
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
041-056_Ch004_RTGO_GEO_12738.indd 44
46°
Divide both sides by 3.
44
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11
) 9 13
Holt Geometry
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201
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10/27/05 7:17:28
45
PM
45
Holt Geometry
Holt Geometry
10/27/05 7:17:29 PM
Ready to Go On? Quiz continued
SECTION
Ready to Go On? Enrichment
SECTION
4A
4A
4-3 Congruent Triangles
Given MNO GHI. Identify the congruent corresponding parts.
_
_
Exploring Exterior Angles
For Exercises 1–4, find the angle measures.
_
_
10. MO GI
11. GH MN
1. m⬔ABD
12. ⬔N H
13. ⬔G M
2. m⬔BDC
A
34
14. LM
3. m⬔BCD
4. m⬔BCE
Given ABC LMN. Find each value.
L
B
33°
(3x)°
C
‹__›
‹__›
‹__› ‹___›
6. m⬔1
Statements
2.
3. UR TS
3.
‹__›
URT RTS
Alt. Int. Thm
13. m⬔QSU
6. RS ⫽ UT
6.
_
RS UT
46⬚
134⬚
_
RT RT
17. m⬔PQR
98⬚
9. Reflex. Prop of 10. 䉭URT 䉭STR
82⬚
360⬚
18. What is the sum of the measures of the exterior angles of the triangle?
s
Def. of 10.
144⬚
16. m⬔SQR
Given
8.
R
(x 2 – 13)° T
15. m⬔SRT
7. Def. segments
8. UR TS
S
36⬚
Given
_
(7x – 3)°
U
14. m⬔QRS
4. Alt Int. ⭄ Thm.
(15x – 7)°
Q
7
12. m⬔QSR
Given
5. Third ⭄ Thm.
9.
For Exercises 11–17, find the indicated values.
11. x ⫽
5. ⬔RUT ⬔RST
_
360⬚
P
Given
1.
2. ⬔SRT ⬔UTR
_
4
T
Reasons
‹__›
_
1
128⬚
10. What is the sum of the measures of the exterior angles of the triangle?
1. RS UT
7.
3
90⬚
U
4.
38°
9. m⬔4
Prove: URT STR
‹___›
360⬚
142⬚
S
Complete the proof.
D
2
52⬚
R
118°
C
7. m⬔2
8. m⬔3
16. Given: RS UT, UR TS, RS ⫽ UT, UR TS
‹__›
E
For Exercises 6–9, find the angle measures.
N
_
_
M
29
5t + 4
47°
5. What is the sum of the measures of the exterior angles of the triangle?
7t – 1
11
15. x
A
B
133⬚
62⬚
71⬚
109⬚
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⬚.
46
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
SECTION
4B
Holt Geometry
Ready to Go On? Skills Intervention
SECTION
4B
4-4 Triangle Congruence: SSS and SAS
041-056_Ch004_RTGO_GEO_12738.indd 46
Holt Geometry
Ready to Go On? Problem Solving Intervention
4-4 Triangle Congruence: SSS and SAS
041-056_Ch004_RTGO_GEO_12738.indd
10/27/05 7:17:30
47
PM
Find these vocabulary words in Lesson 4-4 and the Multilingual Glossary.
10/27/05 7:17:31 PM
Engineers often use triangles in designing structures because of their rigidity.
P
The figure shows a radio tower supported by cables of equal length.
M is the midpoint of LN. Use SSS to explain why 䉭PML 䉭PMN.
Vocabulary
triangle rigidity
47
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
included angle
Understand the Problem
J
Using
_SSS and
_ SAS
_ to Prove Triangles Congruent
_
K
1. Why do you think a radio tower needs to be supported by cables?
JK ML and JK ML. Use SAS to explain why JKM LMK.
_
Sample answer: To protect it from the wind.
_
It is given that JK ML. This means that segment JK
is
parallel
to segment ML. Mark this information on the figure.
_
congruent
_
Alternate
Interior
JKM
Angles
⬔LMK because of the
䉭
LMK
by
SAS
_
A
_
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
SAS
2.
Statements
DB
ML
.
_
MN
.
is congruent to itself.
_
PL ⫽ PN , so PL _
PN by _
the definition of congruent
of a midpoint,
segments. By definition_
_ ML MN . By the Reflexive
Property of Congruence, PM PM . Therefore, 䉭PML 䉭PMN by SSS .
Look Back
8. To use the SSS Theorem to prove triangle congruence, 3 sides of one triangle
Def. of bisector
Reflex. Prop. of 4. SAS
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
7. It is given that
must be congruent to
3.
48
PM
_
.
2.
4. 䉭ABD 䉭CBD
041-056_Ch004_RTGO_GEO_12738.indd 48
Reasons
ABD
CBD
_
_
DB
PL
⫽ PN, and therefore,
Solve
Write a paragraph using the information you found in Exercises 4–6.
Include justifications in your paragraph.
_
1. Given
3. DB
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
DB
6. The segment
.
_ _
1. AB BC , DB bisects ⬔ABC
PL
_
C
_
By the Reflexive Property of Congruence, you know that
Enter this information in Step 3 of the proof.
Therefore, you know that 䉭ABD 䉭CBD by
Enter this information in Step 4 of the proof.
tells you that
B
5. The phrase “M is the midpoint of LN,” tells you that
because of the definition of an angle bisector .
Enter this information in Step 2 of the proof.
_
_
_PN
4. The sentence “The figure shows a tower supported by cables of equal length,”
D
_
_
.
Make a Plan
The 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.
.
.
Proving
_ _ Congruent
_Triangles
Given: AB BC, DB bisects ABC.
Prove: ABD CBD
_
justification
KM
By the Reflexive Property of Congruence, you know that MK JKM
N
3. The problem asks you to “Use SSS to explain why 䉭PML 䉭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
_
Theorem.
_
Therefore, 䉭
L
M
to segment ML. Mark this information on the figure.
_
Since JK ML, you know that ⬔
M
Sample answer: Because of triangle rigidity
_
It is given that JK ML. This means that segment JK
is
L
2. Why do the cables form triangles with the tower and the ground?
3
sides of a second triangle.
9. Have you proven that three sides of 䉭PML are congruent to three sides of
䉭PMN? How? Yes, PM PM , PL PN , ML MN
Holt Geometry
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10/27/05 7:17:32
49
PM
49
Holt Geometry
Holt Geometry
5/25/06 4:29:22 PM
SECTION
Ready to Go On? Skills Intervention
SECTION
4B
4-5 Triangle Congruence: ASA, AAS, and HL
4B
Find this vocabulary word in Lesson 4-5 and the
Multilingual Glossary.
included side
A.
QPR and SRP
According to the diagram,
_QPR and
triangles that share leg
_
_
PR
RP
PR
SRP are
Q
S
P
R
right
.
cannot
hypotenuse
CDE and
B.
_
CE
CE
congruent
of the triangles are
_
CBE are
CE
.
CDE and
right
_
D
CDE
because of the
–
–
, therefore
_
Since AE CD , you know that ⬔
B
CBE
Alternate
E
by HL.
K
J
JKM
_ _
3. AE 储 CD
Given
4. JKM
LMK
Thm
Holt Geometry
4B
coordinate proof
2
x
2
4
/A
.AA
,
-A
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).
x
y
(a,
Substitute the coordinates of L and N into the distance formula to find LN. Simplify.
0) 2 Does LN MO ?
_
So,
LN
a
) (a 0
) 2
a 2
Yes
MO because of the definition of congruent segments.
041-056_Ch004_RTGO_GEO_12738.indd 52
2b
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
–
(9x – 12)°
K
Add 12 to both sides.
Subtract 7x from both sides.
Divide both sides by 2.
8
)4
56
60⬚
4
L(0, 4b)
4b 0
0 2a
N _________, ________
2
2
(a,
)
Does OM ON ?
52
–
2b
0 0
⫺2a 2a, _______
O __________
2
2
(0,
)
0
)
Substitute the coordinates into the Distance Formula and simplify to find OM and ON.
2
2
2
2
OM (x 2 x 1) ( y 2 y 1) ( ⫺a 0) (2b 0) a 2 ⫹ 4b 2
2
2
2
2
2
2
ON (x 2 x 1) ( y 2 y 1) (a 0) ( 2b 0) a ⫹ 4b
_
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Isosceles
Draw 䉭MNO on the diagram above.
a 2
Substitute the coordinates of M and O into the distance formula to find MO. Simplify.
(0 base angle
12
⫺2a 0, ________
0 4b
M _________
2
2
( 2 x 1) 2 ( y 2 1 ) 2
a
m⬔L m⬔K
7x 4 9x ⫺
7x 16 9x
16 2x
x 8
Writing a Coordinate Proof
_ _
Use the square LMNO you drew above to prove that LN ⬵ MO.
(y 2 y 1) base
J
Connect the vertices to form a square.
2
vertex angle
Using Coordinate Proof
M a, 2b)
M(–
N a, 2b)
N(
Given: Isosceles 䉭JKL has coordinates J(⫺2a,
0), K(2a, 0)
_
the
and L(0, 4b)._
M is the midpoint of JL, N is_
midpoint of KL, and O is the midpoint of JK.
J a, 0) O (0, 0) K(2
J(–2
K a, 0)
Prove: 䉭MNO is isosceles.
+ y2
x 1 + x 2 y______
to find the coordinates of M, N, and O.
, 1
Use the Midpoint Formula M ______
2
2
Coordinates of M
Coordinates of N
Coordinates of O
Describe where to place vertex N.
)2 (
10/27/05 7:17:40 PM
(7xx + 4)°
L
Substitute the given values and solve to find x.
m⬔L 7x 4 7(
Move the same distance up from the origin on the y-axis and label this
vertex O(0, a).
0
4-8 Isosceles and Equilateral Triangles
Substitute the value of x into m⬔L and simplify.
Draw another vertex on the x-axis, to the right of origin. Label this
vertex M(a, 0).
(a Ready to Go On? Skills Intervention
From the Isosceles Triangle Theorem, you know that
⬔L ⬔ K . Therefore, m⬔ L m⬔ K .
Use the origin as one vertex of the square. Label it L.
Holt Geometry
Look at the diagram. What type of triangle is 䉭JKL?
Assigning Coordinates to Vertices
Position square LMNO in the coordinate plane and give the
coordinates of each vertex.
Complete and use the distance formula: d 51
Finding the Measure of an Angle
Find m⬔L.
6 from the origin to find the third vertex.
(x 2 x 1)
CPCTC
Alt. Int. ⭄ Thm
legs of an isosceles triangle
Connect the vertices to form a right triangle. Label the vertices with their
coordinates.
MO 6.
spaces to the right to find a second vertex.
2
2
AAS
Vocabulary
y
Use the origin as the vertex of the right angle.
(y 2 y 1) 5.
Find these vocabulary words in Lesson 4-8 and the Multilingual Glossary.
Vocabulary
Positioning a Figure in the Coordinate Plane
Position a right triangle with legs of 7 units and 2 units in
the coordinate plane.
2
2
4. ⬔ A ⬔ D
⬔ E ⬔ C
_
_
6. BE BC
5/26/06 1:44:57
PM
041-056_Ch004_RTGO_GEO_12738.indd
51
Find this vocabulary word in Lesson 4-7 and the
Multilingual Glossary.
(x 2 x 1)
by CPCTC.
_
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
SECTION
4-7 Introduction to Coordinate Proof
LN BC
AAS
Ready to Go On? Skills Intervention
Count 2 units
KM
041-056_CH04_RTGO_GEO_12738.indd 50
up
_
_
BE
and
5. 䉭 ABE 䉭DBC
4.
LMK
50
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
7
AAS
Refl. Prop. Of
3.
Count
Angles Theorem.
_
3. MK
4B
by
C
⬔D and ⬔E ⬔
2. Def. of a Midpt.
Given
Given
SECTION
, because
2. AB DB
1. midpoint of AD
.
L
_
A
_
B is the
_
KM
Complete the flow-chart.
Alt. Int.
DB
Complete the flow-chart:
JKM
LMK
_
Reflexive Property of Congruence, you know that MK
JKM
LMK because of AAS.
Therefore, you know that
2.
_
Fill this information into Steps 5 and 6 of your proof.
L
M
.
Because of the
Given
D
E
.
Interior
DBC
Therefore, 䉭ABE 䉭
J
Mark the given information
on the figure.
_ _
Since it is given that JK ML, you know that
2.
AB
Fill this information into Step 4 of your proof.
Using AAS to Prove
Triangles Congruent
_ _
Given: J
L, JK ML
LMK
Prove: JKM
_ _
1. JK ML
C
B
Fill this information into Step 2 of the proof.
.
_by the Reflexive
_ Property of Congruence.
DE
_
midpoint
of the definition of a
C
BE
It is given that
A
Mark
given_
information on the figure: B is the midpoint of
_ the_
AD and AE CD .
Since B is the midpoint of AD , you know that
be proven by HL. You need to know that the
triangles that share hypotenuse
_
CPCTC
Proving Corresponding Parts
_
_ _Congruent
Given: B
_midpoint of AD; AE 储 CD
_is the
Prove: BE ⬵ BC
_
No
CBE
According to the diagram,
Vocabulary
Fill the given information into Step 1 and Step 3 of the flow-chart proof below.
by the Reflexive Property of Congruence.
Is any information given to you about the hypotenuse of the right triangles?
This conclusion
4-6 Triangle Congruence: CPCTC
Find this vocabulary word in Lesson 4-6 and the
Multilingual Glossary.
Vocabulary
Applying HL Congruence
Determine if you can use the HL Congruence Theorem to prove
the triangles congruent. Explain.
Ready to Go On? Skills Intervention
Yes
Therefore, 䉭MNO is an
Holt Geometry
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203
041-056_Ch004_RTGO_GEO_12738.indd
10/27/05 7:17:43
53
PM
_
Since OM ON, by definition,
isosceles
OM
_
ON .
triangle.
53
Holt Geometry
Holt Geometry
10/27/05 7:17:47 PM
SECTION
Ready to Go On? Quiz
SECTION
4B
Ready to Go On? Quiz continued
4B
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
䉭KIH ⬵ 䉭KIJ.
_
I
H
_
Statements
_
_
U
Q
J
R
V
W
Reasons
_
UV TW , UV TW
2. ⬔VUW
⬔TWU
_ _
3. UW WU
4. 䉭VUW 䉭TWU
1.
U
3. 䉭ABD and 䉭CDB
4-7 Introduction to Coordinate Proof
Position each figure in the coordinate plane.
4. 䉭NMO and 䉭PMO
=
=
D
C
Not enough information.
Yes
10. m⬔Q
2
–2
2O
–2
2
–
_
_
_
2. Def.
Given
_ _
3. ON LM
–
=
48°
D
C
46
of mdpt.
5. 䉭LMK ⬵ 䉭NOK
Ready to Go On? Enrichment
55
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
SECTION
4B
2
MX MY 兹a 2 b 2
Holt Geometry
041-056_CH04_RTGO_GEO_12738.indd 54
2
–2
2O
–2
2
coordinate of X (a, b), coordinate of Y (a, b),
Alt. Int. ⭄ Thm
4.
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: 䉭XMY is isosceles.
AAS
5.
54
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
B
=
(2x – 17)°
Y
Z
M
4. ⬔ L ⬵ ⬔ N
⬔ M ⬵⬔ O
Given
–
N
2. OK ⬵ MK
1.K is mdpt. of OM
A
–
K
O
2
E
11. m⬔E
Q (x + 4)°
L
5. Use AAS to prove the triangles
_congruent.
_ _
Given: K is the midpoint of OM , ON 储 LM
Prove: 䉭LMK ⬵ 䉭NOK
2
4-8 Isosceles and Equilateral Triangles
Find each angle measure.
AD BC or AB CD.
5A
Holt Geometry
Ready to Go On? Skills Intervention
5-1 Perpendicular and Angle Bisectors
041-056_Ch004_RTGO_GEO_12738.indd
5/25/06 4:29:25
55
PM
N
y2 – 8
Trying Triangles
_
1. In the figure at the right, X is the midpoint of AB .
Write a paragraph to explain whether or not
䉭BXM ⬵ 䉭AXN.
X
A
y2 – 5
Vocabulary
equidistant
(15x + 24y)°
M
N
(5x – 4y)°
2. In the figure at the right, ML ⬵ NO , and
m⬔MOL (2x 2). Find ⬔NOM.
Applying the Perpendicular Bisector Theorem and Its Converse
Find each measure.
A. DF
O
L
AB BC CD AD 5;
AC BD 5兹2 ; Since all angles are
y
2
–4 –2 O
–2
–4
B
2
segment
x
points of intersection of the lines are
T
4
2
(4, 4), (3, 5), and (5, 3). The
12x 12x x
2
3
8
4
–4
are 兹 82 , 兹82 , and 兹 128 8兹 2 .
14(
_
=
C
D
B
A
⬔ABD ⬔ADB by Isoc. 䉭 Thm. ⬔ADB ⬔CDB by subst. BD BD
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
V
14x 5
2
x
x
4
)5
51
QR RS because of the
Substitute 46 for RS.
QR by Reflex. Prop. of . 䉭BDC 䉭DBA by ASA.
56
14x – 5
Q
Applying the Angle Bisector Theorem
Find QR.
–
=
–
⭄ Thm. ⬔CDB ⬔CBD by Isoc. 䉭 Thm. ⬔ABD ⬔CBD by subst.
U 艎
W
Substitute the value of x to find VU.
14x 5
Sample answer: Yes. Given CD BA, ⬔CDB ⬔ABD by Alt. Int.
12x + 3
5 14x 5 5 12x
4
lengths of the sides of the triangles
5. In the figure at right, CD 储 BA. Is this enough information to show
that 䉭BDC ⬵ 䉭DBA? Explain your reasoning.
E
.
Bisector
TU VU because of the Perpendicular
Theorem. Substitute the given measures for TU and VU and
solve for x.
12x 3 –4 –2 O
–2
3
3
DF C
y
This is an isosceles triangle. The
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
bisector
B. VU
4. What kind of triangle is formed by the lines
y 9x 32, x y 2, and x 9y 32?
Explain your answer.
5.8
Substitute 3 for FE.
4
lengths, then ABCD is a square.
_
of DE by the Converse of the
艎
F
C
Perpendicular
Bisector
Theorem.
Therefore, DF FE because of the definition of a
A
4
D
5.8
perpendicular
_
bisector
90 and all sides have the same
D
_ ‹___›
Since CD CE, and ᐉ ⬜ DE , CF is the
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.
focus
M
_
38
10/27/05 7:17:56 PM
Find these vocabulary words in Lesson 5-1 and the Multilingual Glossary.
4yy + 3
XA XB 31, XM 27 but XN 28.
_
6yy – 5
B
The triangles are not congruent. y 6,
041-056_CH04_RTGO_GEO_12738.indd 56
–2
2O
–2
2
M
_ _
2
2
Check student’s graph and verify distance formula
calculations.
P
Need to know that
SECTION
8. a right triangle with legs
5 units in length.
2
9. Assign coordinates to each vertex and write a coordinate proof
Given: rectangle WXYZ
Prove: WX ⬵ YZ
O
N
Check student’s graphs
7. a square with length 5 units
–2
2O
–2
2
B
Reasons
Given
2. Alt. Int. ⭄ Thm
3. Vert ⭄ Thm
4. AAS
5. CPCTC
1.
S
Given
2. Alt. Int. ⭄ Thm
3. Reflex. Prop. Of 4. SAS
1.
4-5 Triangle Congruence ASA, AAS, and HL
Determine if you can use the HL Congruence Theorem to prove the
triangles congruent. If not, tell what else you need to know.
A
Statements
TU RS; TU RS
2. ⬔T ⬔S
3. ⬔TQU ⬔SQR
4. 䉭TQU 䉭SQR
5. QS QT
1.
_
2. Given: UV ⬵ TW , UV 储 TW
Prove : 䉭VUW ⬵ 䉭TWU
T
_ _
_
it is given that KH KJ: KI KI by the Reflexive
Property of Congruence. So 䉭KIH 䉭KIJ by SAS.
_ _
Triangle Congruence CPCTC
, TU ⬵ RS
6. Given TU
_
_储 RS
T
Prove: QS ⬵ QT
_
Sample answer:_
Since_
KI _
bisects_⬔HKJ, ⬔HKI ⬔JKI ;
_
4-6
K
Holt Geometry
R
Bisector
46
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
204
Angle
057-075_Ch005_RTGO_GEO_12738.indd
5/26/06 2:37:44
57
PM
Theorem.
27°
27°
P
57
艎
46
S
Holt Geometry
Holt Geometry
11/1/05 7:22:32 PM