Skills Practice
Skills Practice for Lesson 1.1
Name _____________________________________________
Date ____________________
Let’s Get This Started
Points, Lines, Planes, Rays, and Segments
Vocabulary
1
Write the term that best completes each statement.
1. A geometric figure created without using tools is a(n)
are two or more lines that are not in the same plane.
2.
3. A(n)
.
is a location in space.
4. The points where a line segment begins and ends are its
5. A(n)
of points.
.
is a straight continuous arrangement of an infinite number
6. Two or more line segments of equal measure are
.
7. You
a geometric figure when you use only a compass
and straightedge.
8. Points that are all located on the same line are
.
© 2010 Carnegie Learning, Inc.
9. A(n)
is a portion of a line that includes two points and all
of the collinear points between the two points.
10. You can
exact copy of the original line segment.
11. A flat surface is a(n)
by constructing an
.
12. A(n)
is a portion of a line that begins with a single point and extends
infinitely in one direction.
13. Two or more lines located in the same plane are
.
14. When you
a geometric figure, you use tools such as a ruler,
straightedge, compass, or protractor.
Chapter 1 ● Skills Practice
281
Problem Set
Identify the point(s), line(s), and plane(s) in each figure.
1.
Points: A, B, and C
A
___›
C
‹
‹___›
Lines: AB and BC
B
1
Plane: m
m
2.
Z
X
Y
a
3.
p
R
S
4.
L
M
N
x
282
Chapter 1 ● Skills Practice
© 2010 Carnegie Learning, Inc.
Q
Name _____________________________________________
Date ____________________
Draw a figure for each description. Label all points mentioned
in the description.
5. Points R, S, and T are collinear such that point T is located halfway between
points S and R.
T
S
R
1
6. Points A, D, and X are collinear such that point A is located halfway between
points D and X.
7. Points A, B, and C are collinear such that point B is between points A and C and
the distance between points A and B is twice the distance between points B and C.
8. Points F, G, and H are collinear such that point F is between points G and H
and the distance between points F and G is one third the distance between
points G and H.
Identify all examples of coplanar lines in each figure.
© 2010 Carnegie Learning, Inc.
9.
10.
m
p
c
q
d
n
a
b
Lines m and p are coplanar.
Lines n and q are coplanar.
Chapter 1 ● Skills Practice
283
11.
w
z
12.
q
x
p
y
r
u
t
s
1
Identify all skew lines in each figure.
13.
14.
g
h
a
c
f
b
Lines f and g are skew.
Lines f and h are skew.
16.
n
w
m
x
y
284
Chapter 1 ● Skills Practice
l
© 2010 Carnegie Learning, Inc.
15.
Name _____________________________________________
Date ____________________
Draw and label an example of each geometric figure.
‹___›
___
17. XY
X
18. CD
Y
1
___›
___
‹
19. PR
20. FG
‹____›
___›
21. HM
22. KJ
Use symbols to write the name of each geometric figure.
© 2010 Carnegie Learning, Inc.
23.
R
T
24.
A
B
___
RT
25.
26.
X
M
N
Y
Chapter 1 ● Skills Practice
285
27.
28.
C
S
D
R
Use a ruler to measure each segment to the nearest centimeter. Then use
symbols to write an equation that expresses the measure of each segment.
29.
A
B
___
AB 4 centimeters or m AB 4 centimeters
30.
31.
32.
A
B
B
A
B
A
Use a compass and a straightedge to copy each line segment. Then write a
congruence statement to show that the segments have the same length.
33.
P
A
___ ___
C
B
PC ⬵ AB
34.
286
X
Y
Chapter 1 ● Skills Practice
© 2010 Carnegie Learning, Inc.
1
Name _____________________________________________
35.
Date ____________________
S
R
1
36.
M
N
Use each set of line segments to construct a line segment with the
indicated length.
© 2010 Carnegie Learning, Inc.
37. Construct a line segment with a length equal to AB ST.
B
A
S
S
A
T
T
B
38. Construct a line segment with a length equal to 3MN.
M
N
Chapter 1 ● Skills Practice
287
39. Construct a line segment with a length equal to 2CD GH.
G
C
H
D
1
40. Construct a line segment with a length equal to PQ 3JK.
P
K
J
© 2010 Carnegie Learning, Inc.
Q
288
Chapter 1 ● Skills Practice
Skills Practice
Skills Practice for Lesson 1.2
Name _____________________________________________
Date ____________________
All About Angles
Naming Angles, Classifying Angles, Duplicating Angles, and
Bisecting Angles
1
Vocabulary
Match each term to its definition.
1. bisect
a. an angle whose measure is equal to 90°
2. angle
b. to construct an exact copy of an angle
3. right angle
c. an angle whose measure is greater than
90° but less than 180°
4. vertex of an angle
5. degrees
6. duplicate an angle
7. protractor
e. the two rays that form an angle
8. acute angle
f. the common endpoint of the two rays that
form an angle
9. obtuse angle
g. to divide into two equal parts
10. sides of an angle
11. congruent angles
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d. a ray that divides an angle into two
congruent angles
h. a unit of measure for angles
12. angle bisector
i. an angle whose measure is greater than 0°
but less than 90°
13. straight angle
j. a basic tool used to measure angles
k. two or more angles that have
equal measures
l. an angle whose measure is equal to 180°
m. a geometric figure formed by two rays that
share a common endpoint
Chapter 1 ● Skills Practice
289
Problem Set
Name the vertex and sides of each angle.
1.
2.
P
B
Q
R
A
1
The vertex is point R.
___›
C
____›
The sides are RP and RQ.
3.
W
4.
Y
H
X
G
F
5. 2
6. 1
G
F
A
1
2
K
J
FKJ or JKF
290
Chapter 1 ● Skills Practice
1
D 2
C
B
© 2010 Carnegie Learning, Inc.
Write two alternate names for each angle.
Name _____________________________________________
7. ⬔LNM
Date ____________________
8. ⬔YZX
X
L
M
2
1
1 2
N
W
Z
Y
K
1
Use the diagram to determine the measure of each angle to the
nearest degree.
90
100
80
110
70
P
12
0
60
13
0
50
0
15
30
0
15
30
14
0
50
0
13
80
70
100
60
110
0
12
0
14
40
40
9. m⬔PDX ⫽ 40°
170
10
10
20
170
160
160
20
B
D
X
10. m⬔RUS ⫽
10
20
170
160
30
15
0
40
14
0
80
70
100
60
110
0
12
50
0
13
U
R
90
80
100
110
70
1
2
0
60
13
0
50
0
14
40
30
15
0
160
20
170
10
© 2010 Carnegie Learning, Inc.
S
T
Chapter 1 l Skills Practice
291
110
70
120
130
60
50
140
40
15
0
30
16
0
20
W
1
B
30
40
20
50
10
160 150 140 13
0
170
60
12
0
80
0
10
90
100
80
0
17
10
70
11
0
11. mWOB O
P
12. mTAL T
80
10
0
L
90
10
0
17
50
40
60
30
70
140 130 120
20
150
110
160
80
0
10
110
70
120
60
130
50
140
40
A
150
30
160
20
C
13. mCGM 122°
C
G
292
Chapter 1 ● Skills Practice
M
© 2010 Carnegie Learning, Inc.
17
0
10
Use a protractor to determine the measure of each angle to the
nearest degree.
Name _____________________________________________
Date ____________________
14. mBPX B
P
X
1
15. mTZJ T
J
Z
16. mCAV C
V
© 2010 Carnegie Learning, Inc.
A
Use a protractor to draw an angle with each measure.
17. 28°
18. 112°
Chapter 1 ● Skills Practice
293
19. 90°
20. 180°
1
Determine whether each angle is an acute angle, an obtuse angle, a right
angle, or a straight angle.
21. mACD 96°
22. mVHG 180°
obtuse angle
23. mTUX 68°
24. mKOP 90°
Use a compass and a straightedge to copy each angle. Then write a
congruence statement to show that the angles have the same measure.
C
B
26.
D
D
S
R
T
CBD ⬵ SRT
294
Chapter 1 ● Skills Practice
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25.
Name _____________________________________________
27.
Date ____________________
28.
P
Z
1
Construct the angle bisector of each angle.
29.
30.
A
D
P
B
© 2010 Carnegie Learning, Inc.
B
31.
32.
S
X
Chapter 1 ● Skills Practice
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© 2010 Carnegie Learning, Inc.
1
296
Chapter 1 ● Skills Practice
Skills Practice
Skills Practice for Lesson 1.3
Name _____________________________________________
Date ____________________
Special Angles
Complements, Supplements, Midpoints, Perpendiculars,
and Perpendicular Bisectors
1
Vocabulary
© 2010 Carnegie Learning, Inc.
Draw a figure to illustrate each term.
1. supplementary angles
2. complementary angles
3. adjacent angles
4. perpendicular bisector
5. midpoint of a segment
6. perpendicular
7. linear pair
8. vertical angles
Chapter 1 ● Skills Practice
297
Problem Set
Use a protractor to draw an angle that is supplementary to each given
angle. Draw the angle so it shares a common side with the given angle.
Label the measure of each angle.
1.
2.
1
135° 45°
4.
Use a protractor to draw an angle that is supplementary to each given
angle. Draw the angle so it does not share a common side with the given
angle. Label the measure of each angle.
5.
6.
122°
298
58°
Chapter 1 ● Skills Practice
© 2010 Carnegie Learning, Inc.
3.
Name _____________________________________________
7.
Date ____________________
8.
1
Use a protractor to draw an angle that is complementary to each given
angle. Draw the angle so it shares a common side with the given angle.
Label the measure of each angle.
9.
28°
10.
62°
© 2010 Carnegie Learning, Inc.
11.
12.
Use a protractor to draw an angle that is complementary to each given
angle. Draw the angle so it does not share a common side with the given
angle. Label the measure of each angle.
13.
79°
11°
14.
Chapter 1 ● Skills Practice
299
15.
16.
19°
36°
Solve for x.
17.
18.
x
34°
107°
x
x 180° 107° 73°
19.
20.
124°
x
x
58°
21.
22.
6°
x
x 63°
300
Chapter 1 ● Skills Practice
© 2010 Carnegie Learning, Inc.
1
Name _____________________________________________
Date ____________________
Use the given information to determine the measure of each pair of angles.
23. The measure of the complement of an angle is three times the measure of the
angle. What is the measure of each angle?
x 3x 90
4x 90
x 22.5
1
The measure of the angle is 22.5° and the measure of the complement is 67.5°.
24. The measure of the supplement of an angle is one fourth the measure of the angle.
What is the measure of each angle?
© 2010 Carnegie Learning, Inc.
25. The measure of the supplement of an angle is twice the measure of the angle.
What is the measure of each angle?
26. The measure of the complement of an angle is one fifth the measure of the angle.
What is the measure of each angle?
Chapter 1 ● Skills Practice
301
Construct each perpendicular line described.
27. Construct a line that is perpendicular to line CD and passes through point T.
1
T
C
D
28. Construct a line that is perpendicular to line AB and passes through point X.
B
29. Construct a line that is perpendicular to line MN and passes through point J.
J
N
M
302
Chapter 1 ● Skills Practice
© 2010 Carnegie Learning, Inc.
X
A
Name _____________________________________________
Date ____________________
30. Construct a line that is perpendicular to line PQ and passes through point R.
P
1
R
Q
Construct a perpendicular bisector through each segment and label
the midpoint M.
31.
© 2010 Carnegie Learning, Inc.
V
M
J
32.
S
P
Chapter 1 ● Skills Practice
303
33.
C
1
D
34.
K
R
Determine whether angles 1 and 2 are adjacent angles.
35.
36.
2
2
The angles are not adjacent.
37.
38.
1
2
304
Chapter 1 ● Skills Practice
1
2
© 2010 Carnegie Learning, Inc.
1
1
Name _____________________________________________
Date ____________________
Determine whether angles 1 and 2 form a linear pair.
39.
40.
1
2
1
2
1
The angles do not form a linear pair.
41.
42.
2
1
1
2
Name each pair of vertical angles.
43.
44.
1
© 2010 Carnegie Learning, Inc.
5
2 6
3 7
4 8
2
10
9
12
1 5
6
11
4
3 7
8
1 and 6, 2 and 5, 3 and 8,
4 and 7, 9 and 11, 10 and 12
45.
46.
1 9
2 10
11
3
1
2 5
6
8
5 6
7
3 7
4 8
10 9
11 12
12
4
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1
306
Chapter 1 ● Skills Practice
Skills Practice
Name _____________________________________________
Skills Practice for Lesson 1.4
Date ____________________
A Little Dash of Logic
Two Methods of Logical Reasoning
Vocabulary
1
Define each term in your own words.
1. inductive reasoning
2. deductive reasoning
Problem Set
© 2010 Carnegie Learning, Inc.
For each situation, identify the specific information, the general information,
and the conclusion.
1. You read an article in the paper that says a high-fat diet increases a person’s risk
of heart disease. You know your father has a lot of fat in his diet, so you worry that
he is at great risk of heart disease.
Specific information: Your father has a lot of fat in his diet.
General information: High-fat diets increase the risk of heart disease.
Conclusion: Your father is at higher risk of heart disease.
2. You hear from your teacher that spending too much time in the sun without
sunblock increases the risk of skin cancer. Your friend Susan spends as much time
as she can outside working on her tan without sunscreen, so you tell her that she is
increasing her risk of skin cancer when she is older.
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3. Janice tells you that she has been to the mall three times in the past week, and every
time there were a lot of people there. “It’s always crowded at the mall,” she says.
4. John returns from a trip out West and reports that it was over 100 degrees
every day. “It’s always hot out West,” he says.
Determine the type of reasoning used in each situation. Then determine
whether the conclusion is correct.
5. Jason sees a line of 10 school buses and notices that each is yellow. He concludes
that all school buses must be yellow. What type of reasoning is this? Is his
conclusion correct? Explain.
It is inductive reasoning because he has observed specific examples of a
phenomenon—the color of school buses—and come up with a general rule
based on those specific examples.
The conclusion is not necessarily true. It may be the case, for example, that
all or most of the school buses in this school district are yellow, while another
school district may have orange school buses.
6. Caitlyn has been told that every taxi in New York City is yellow. When she sees
a red car in New York City, she concludes that it cannot be a taxi. What type of
reasoning is this? Is her conclusion correct? Explain.
308
Chapter 1 ● Skills Practice
© 2010 Carnegie Learning, Inc.
1
Name _____________________________________________
Date ____________________
7. Miriam has been told that lightning never strikes twice in the same place. During
a lightning storm, she sees a tree struck by lightning and goes to stand next to it,
convinced that it is the safest place to be. What type of reasoning is this? Is her
conclusion correct? Explain.
1
8. Jose is shown the first six numbers of a series of numbers: 7, 11, 15, 19, 23, 27.
He concludes that the general rule for the series of numbers is a n 4n 3.
What type of reasoning is this? Is his conclusion correct? Explain.
Write a paragraph for each question.
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9. Provide your own example of deductive reasoning. Explain your answer.
The main feature to look for, both in the example and the explanation,
is the idea of working from a general rule to draw a conclusion about a
specific situation.
For example: Kelly has been told that being overweight increases a person’s
chance of becoming diabetic. Noticing that her uncle is overweight, she tells
him that he should have regular checkups to look out for diabetes. This is
deductive reasoning because she concludes from the general rule about
overweight people that her uncle has a higher-than-normal risk of
becoming diabetic.
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10. Provide your own example of inductive reasoning. Explain your answer.
11. Write a brief paragraph explaining what inductive reasoning is, as if you are telling
your parents what you learned in school, and offer an example.
12. Write a brief paragraph explaining what deductive reasoning is, as if you are telling
your parents what you learned in school, and offer an example.
In each situation, identify the type of reasoning that each of the two people
are using. Then compare and contrast the two types of reasoning.
13. When Madison babysat for the Johnsons for the first time, she was there 2 hours and
was paid $30. The next time she was there for 5-hours and was paid $75. She decided
that the Johnsons were paying her $15 per hour. The third time she went, she stayed
for 4 hours. She tells her friend Jennifer that she makes $15 per hour babysitting. So,
Jennifer predicted that Madison made $60 for her 4-hour babysitting job.
Madison used inductive reasoning to conclude that the Johnsons were paying
her at a rate of $15 per hour. From that general rule, Jennifer used deductive
reasoning to conclude that 4 hours of babysitting should result in a payment
of $60. Inductive reasoning looks at evidence and creates a general rule from
the evidence. By contrast, deductive reasoning starts with a general rule
and makes a prediction or deduction about what will happen in a
particular instance.
310
Chapter 1 ● Skills Practice
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1
Name _____________________________________________
Date ____________________
14. When Holly was young, the only birds she ever saw were black crows. So, she told
her little brother Walter that all birds are black. When Walter saw a bluebird for the
first time, he was sure it had to be something other than a bird.
1
© 2010 Carnegie Learning, Inc.
15. Tamika is flipping a coin and recording the results. She records the following
results: heads, tails, heads, tails, heads, tails, heads. She tells her friend Javon that
the coin alternates between heads and tails for each toss. Javon tells her that the
next time the coin is flipped, it will definitely be tails.
16. John likes to watch the long coal trains moving past his house. Over the weeks
of watching he notices that every train going east is filled with coal, but the trains
heading west are all empty. He tells his friend Richard that all trains heading east
have coal and all trains heading west are empty. When Richard hears a train
coming from the west, he concludes that it will certainly be filled with coal.
Chapter 1 ● Skills Practice
311
© 2010 Carnegie Learning, Inc.
1
312
Chapter 1 ● Skills Practice
Skills Practice
Name _____________________________________________
Skills Practice for Lesson 1.5
Date ____________________
Conditionals
Conditional Statements, Postulates, and Theorems
Vocabulary
1
Define each term in your own words.
1. conditional statement
2. propositional form
3. hypothesis
4. conclusion
5. propositional variables
© 2010 Carnegie Learning, Inc.
6. truth value
7. postulate
8. theorem
9. Euclidean geometry
10. hyperbolic geometry
11. elliptic geometry
Chapter 1 ● Skills Practice
313
State each postulate.
12. Linear Pair Postulate
13. Segment Addition Postulate
14. Angle Addition Postulate
Problem Set
Write each statement in propositional form.
1. The measure of an angle is 90°. So, the angle is a right angle.
If the measure of an angle is 90°, then the angle is a right angle.
2. Three points are all located on the same line. So, the points are collinear points.
3. Two lines are not on the same plane. So, the lines are skew.
4. Two angles are supplementary angles if the sum of their angle measures is equal
to 180°.
5. Two angles share a common vertex and a common side. So, the angles are
adjacent angles.
6. A ray divides an angle into two congruent angles. So, the ray is an angle bisector.
314
Chapter 1 ● Skills Practice
© 2010 Carnegie Learning, Inc.
1
Name _____________________________________________
Date ____________________
Identify the hypothesis and the conclusion of each conditional statement.
7. If two lines intersect at right angles, then the lines are perpendicular.
The hypothesis is “Two lines intersect at right angles.”
The conclusion is “The lines are perpendicular.”
1
8. If the sum of two angles is 180º, then the angles are supplementary.
9. If the sum of two adjacent angles is 180º, then the angles form a linear pair.
10. If the measure of an angle is 180°, then the angle is a straight angle.
© 2010 Carnegie Learning, Inc.
11. If two lines are located in the same plane, then the lines are coplanar lines.
12. If the sum of two angle measures is equal to 90°, then the angles are
complementary angles.
Chapter 1 ● Skills Practice
315
Answer each question about the given conditional statement.
13. Conditional statement: If the measure of angle ABC is 45 degrees and the measure
of angle XYZ is 45 degrees, then ABC XYZ.
What does it mean if the hypothesis is false and the conclusion is true, and then
what is the truth value of the conditional statement?
If the hypothesis is false and the conclusion is true, then the measure of angle ABC is
not 45 degrees and the measure of angle XYZ is not 45 degrees, and angles ABC and
XYZ are congruent. The truth value of the conditional statement is true, because the
angles could have measures that are equal, but different than 45 degrees.
1
14. Conditional statement: If the measure of angle XYZ is less than 90 degrees,
then angle XYZ is acute.
What does it mean if the hypothesis is true and the conclusion is false, and then
what is the truth value of the conditional statement?
15. Conditional statement: If 1 and 2 are two nonadjacent angles formed by two
intersecting lines, then they are vertical angles.
16. Conditional statement: If the measure of LMN is 180°, then LMN is a straight angle.
What does it mean if the hypothesis is false and the conclusion is false, and then
what is the truth value of the conditional statement?
316
Chapter 1 ● Skills Practice
© 2010 Carnegie Learning, Inc.
What does it mean if the hypothesis is true and the conclusion is true, and then
what is the truth value of the conditional statement?
Name _____________________________________________
Date ____________________
Draw each scenario. Then rewrite each conditional statement by separating
the hypothesis and conclusion.
___›
17. If RT bisects PRS, then PRT and SRT are adjacent angles.
P
R
T
1
S
___›
Given: RT bisects PRS
Prove: PRT and SRT are adjacent angles
18. If QRS and SRT are complementary angles, then mQRS mSRT 90°.
Given:
© 2010 Carnegie Learning, Inc.
Prove:
‹___›
___
‹___›
___
‹___›
___
19. If AB KJ and AB bisects KJ , then AB is the perpendicular bisector of KJ .
Given:
Prove:
Chapter 1 ● Skills Practice
317
___›
20. If PG bisects FPH, then FPG 艑 GPH.
1
Given:
Prove:
Write the postulate that confirms each statement.
___
21. Angles GFH and KFH are
supplementary angles.
G
H
___
___
22. mRS mST mRT
Q
R
S
T
F
J
K
Linear Pair Postulate
Y
24. m1 m2 180°
X
V
Z
318
W
Chapter 1 ● Skills Practice
2 1
3 45
© 2010 Carnegie Learning, Inc.
23. mWXZ mZXY mWXY
Name _____________________________________________
25. BC CD BD
Date ____________________
26. mDBE mEBF mDBF
A
B
E
D
C
F
D
E
C
B
A
1
Complete each statement. The write the postulate you used.
____
____
___
K
28. mAB m
L M
N
___
___
27. mLM mMN mLN
P
B
A
mAC
D
C
E
Segment Addition Postulate
29. mYVZ m
180°
m
30. m
K
J
Y
Z
X
© 2010 Carnegie Learning, Inc.
mMJK
L
V
M
W
Chapter 1 ● Skills Practice
319
___
mGI m
31. m
F
G
mRPS mRPT
32. m
H
I
R
S
T
Q
P
© 2010 Carnegie Learning, Inc.
1
320
Chapter 1 ● Skills Practice
Skills Practice
Skills Practice for Lesson 1.6
Name _____________________________________________
Date ____________________
Forms of Proof
Paragraph Proof, Two-Column Proof, Construction Proof, and
Flow Chart Proof
1
Vocabulary
Match each definition to its corresponding term.
1. If a is a real number, then a a.
a. Addition Property of Equality
2. If a, b and c are real numbers, a b,
and b c, then a c.
b. paragraph proof
3. If a, b, and c are real numbers and
a b, then a c b c.
4. a proof in which the steps and corresponding
reasons are written in complete sentences
5. If a and b are real numbers and a b,
then a can be substituted for b.
© 2010 Carnegie Learning, Inc.
6. a proof in which the steps are written in the
left column and the corresponding reasons
in the right column
c. construction proof
d. Subtraction Property of Equality
e. Transitive Property
f. flow chart proof
g. Substitution Property
h. two-column proof
i. Reflexive Property
7. a proof in which the steps and
corresponding reasons are written in boxes
8. If a, b, and c are real numbers and a b,
then a c b c.
9. a proof that results from creating an object
with specific properties using only a
compass and straightedge
Chapter 1 ● Skills Practice
321
Problem Set
1. mABC mXYZ
mABC mRST mXYZ mRST
___
___
2. m QT
m TU
___
____
___
____
m QT m WX mTU mWX
Subtraction Property of Equality
4. GH MN and MN OP,
so GH OP
1
___
___
___
___
5. mXY ___
4 cm___
and mBC 4 cm,
so mXY mBC
6. PR ⬵ PR
7. GH JK
GH RS JK RS
8. m1 134° and m2 134°,
so m1 m2
9. mABC mDEF
ABC mQRS mDEF mQRS
11. ED 3 in. and PQ 3 in., so
ED PQ
10. GH GH
12. EFG ⬵ LMN and LMN ⬵ SPT,
so EFG ⬵ SPT
Write a statement that fits the given description.
13. Write a segment statement using the Reflexive Property.
___ ___
Sample Answer: XY ⬵ XY
14. Write angle statements using the Addition Property of Equality.
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Chapter 1 ● Skills Practice
© 2010 Carnegie Learning, Inc.
3. JKL ⬵ JKL
Name _____________________________________________
Date ____________________
15. Write angle statements using the Substitution Property.
16. Write segment statements using the Transitive Property.
1
17. Write segment statements using the Subtraction Property of Equality.
18. Write an angle statement using the Reflexive Property.
Rewrite each conditional statement by separating the hypothesis and
conclusion. The hypothesis becomes the “Given” information and the
conclusion becomes the “Prove” information.
19. Conditional statement: If 2 ⬵ 1, then 2 ⬵ 3.
© 2010 Carnegie Learning, Inc.
Given: 2 ⬵ 1
Prove: 2 ⬵ 3
___
___
___
___
20. Conditional statement: RT ⬵ LM , if RT ⬵ AB
Given:
Prove:
21. Conditional statement: if mABC mLMN then mABC mXYZ
Given:
Prove:
22. Conditional statement: AB RS CD RS, if AB CD
Given:
Prove:
Chapter 1 ● Skills Practice
323
Use the indicated form of proof to prove each statement.
23. Prove the following using a two-column proof.
___
___
___
___
___
___
D
Given: mAX mCX
X
Given: mBX mDX
Prove: mAB mCD
1
1.
2.
3.
4.
5.
6.
B
A
C
___Statements
___
mAX mCX
___
___
mBX mDX
___
___
___
___
mAX mBX mCX mBX
___
___
___
___
mAX mBX mCX mDX
___
___
___
mAX mBX mAB
____
___
___
mCX mDX mCD
___
___
7. mAB mCD
Reasons
1. Given
2. Given
3. Addition Property of Equality
4. Substitution Property
5. Segment Addition property
6. Segment Addition Postulate
7. Substitution Property
24. Prove the following using a construction proof.
____
___
___
____
Given: KM ⬵ LN
Prove: KL ⬵ MN
M N
© 2010 Carnegie Learning, Inc.
K L
324
Chapter 1 ● Skills Practice
Name _____________________________________________
Date ____________________
25. Prove the following using a simple paragraph proof.
Given: VZW ⬵ XZY
Prove: VZX ⬵ WZY
V
W
X
Z
1
Y
26. Prove the following using a flow chart proof.
Given: ABC and XYZ are straight angles.
© 2010 Carnegie Learning, Inc.
Prove: ABC ⬵ XYZ
Chapter 1 ● Skills Practice
325
27. Prove the following using a simple paragraph proof.
Given: A is supplementary to B
Given: C is supplementary to D
Given: A ⬵ D
Prove: B ⬵ C
1
28. Prove the following using a two-column proof.
‹___›
‹___›
Given: AB DE
Prove: ABD ⬵ CBD
D
A
B
C
Statements
326
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
Chapter 1 ● Skills Practice
© 2010 Carnegie Learning, Inc.
E
Name _____________________________________________
Date ____________________
Write each given proof as the indicated proof.
29. Write the flow chart proof below as a two-column proof.
Given: PQT ⬵ RQS
S
Prove: PQS ⬵ RQT
T
P
© 2010 Carnegie Learning, Inc.
∠PQT ≅ ∠RQS
Given
m∠PQT = m∠RQS
Definition of
congruent angles
Q
m∠SQT = m∠SQT
Identity
m∠PQS + m∠SQT =
m∠RQT + m∠SQT
Substitution
m∠PQT = m∠PQS + m∠SQT
Angle Addition Postulate
m∠PQS = m∠RQT
Subtraction Property
of Equality
m∠RQS = m∠RQT + m∠SQT
Angle Addition Postulate
∠PQS ≅ ∠RQT
Definition of
congruent angles
Statements
1
R
Reasons
1. PQT ⬵ RQS
1. Given
2. mPQT mRQS
2. Definition of congruent angles
3. mPQT mPQS mSQT
3. Angle Addition Postulate
4. mRQS mRQT mSQT
4. Angle Addition Postulate
5. mPQS mSQT mRQT mSQT
5. Substitution
6. mSQT mSQT
6. Identity
7. mPQS mRQT
7. Subtraction Property of Equality
8. PQS ⬵ mRQT
8. Definition of congruent angles
Chapter 1 ● Skills Practice
327
30. Write the flow chart proof of the Right Angle Congruence Theorem below as a
two-column proof.
Given: Angles ACD and BCD are right angles.
Prove: ACD ⬵ BCD
D
A
1
C
ACD is a right angle
BCD is a right angle
Given
Given
mACD 90°
mBCD 90°
Definition of right angles
Definition of right angles
B
m
BCD
90°
ACD
≅ BCD
Definition
of right angles
Definition
of congruent
angles
328
Chapter 1 ● Skills Practice
© 2010 Carnegie Learning, Inc.
mACD
mBCD
mBCD
90°
Transitive
Property
Equality
Definition
of rightofangles
Name _____________________________________________
Date ____________________
31. Write the two-column proof of the Congruent Supplement Theorem below as a
paragraph proof.
Given: 1 is supplementary to 2,
3 is supplementary to 4,
and 2 ⬵ 4
4
1
3
2
Prove: 1 ⬵ 3
Statements
1. 1 is supplementary to 2
1. Given
2. 3 is supplementary to 4
2. Given
3. 2 ⬵ 4
3. Given
4. m2 m4
4. Definition of congruent angles
5. m1 m2 180°
5. Definition of supplementary angles
6. m3 m4 180°
6. Definition of supplementary angles
7. m1 m2 m3 m4
7. Substitution Property
8. m1 m2 m3 m2
8. Substitution Property
9. m1 m3
9. Subtraction Property of Equality
10. Definition of congruent angles
© 2010 Carnegie Learning, Inc.
10. 1 ⬵ 3
1
Reasons
Chapter 1 ● Skills Practice
329
32. Write the two-column proof of the Congruent Complement Theorem below as a
paragraph proof.
Given: Angles ABD and DBC are
complementary, angles WXZ and ZXY are
complementary, and DBC ⬵ ZXY
Prove: ABD ⬵ WXZ
1
Y
A
Z
B
Statements
X
D
W
C
Reasons
1. ABD is complementary to DBC
1. Given
2. WXZ is complementary to ZXY
2. Given
3. DBC ⬵ ZXY
3. Given
4. mABD mDBC 180°
4. Definition of complementary angles
5. mWXZ mZXY 180°
5. Definition of complementary angles
6. mDBC mZXY
6. Definition of congruent angles
7. mABD mDBC
mWXZ mZXY
7. Substitution Property
8. mABD mDBC
mWXZ mDBC
8. Substitution Property
9. mABD mWXZ
9. Subtraction Property of Equality
10. Definition of congruent angles
© 2010 Carnegie Learning, Inc.
10. ABD ⬵ WXZ
330
Chapter 1 ● Skills Practice
Name _____________________________________________
Date ____________________
33. Write the paragraph proof below as a flow chart proof.
Given: mQXR mSXR
Prove: mPXR mTXR
R
S
Q
X
P
T
© 2010 Carnegie Learning, Inc.
By the Angle Addition Postulate, mTXR mTXS mSXR. It is given that
mQXR mSXR, so by substitution, mTXR mTXS mQXR. Angles PXQ
and TXS are vertical angles by the definition of vertical angles. Vertical angles are
congruent by the Vertical Angle Theorem, so PXQ ⬵ TXS, and by the definition
of congruent angles, mPXQ mTXS. Using substitution, you can write
mTXR mPXQ mQXR. By the Angle Addition Postulate,
mPXR mPXQ mQXR. So, you can use substitution to write mPXR mTXR.
Chapter 1 ● Skills Practice
331
1
34. Write the paragraph proof below as a flow chart proof.
____
___
___
___
___
___
Given: GH ⬵ HJ and FH ⬵ HK
J
G
Prove: GK ⬵ FJ
H
K
F
____
___
By the Segment Addition Postulate, GK GH HK. You are given that GH HJ ,
so GH HJ by the definition of congruent segments,
and
you can use substitution
___
___
to write GK HJ HK. You are also given that FH ⬵ HK , so FH HK by the
definition of congruent segments, and you can use substitution to write GK HJ FH. By the Segment Addition Postulate, FJ FH HJ. So, you can
use___
___
substitution to write GK FJ. By the definition of congruent segments, GK ⬵ FJ .
© 2010 Carnegie Learning, Inc.
1
332
Chapter 1 ● Skills Practice