transversals and lines angles formed by …...2011/11/07 · conjectures about angles formed by...
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Chapter 2 ● Skills Practice 333
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Skills Practice Skills Practice for Lesson 2.1
Name _____________________________________________ Date ____________________
Transversals and LinesAngles Formed by Transversals of Parallel and Non-Parallel Lines
VocabularyWrite the term that best completes each statement. Use the figure for Exercises 4 – 10.
alternate interior parallel skew
alternate exterior transversal interior
same side interior corresponding exterior
same side exterior
1. Coplanar lines that never intersect are called lines.
2. Non-coplanar lines are called lines.
3. A line that intersects two or more lines at distinct points is called
a(n) .
4. Angles 3 and 4 are angles because they
are on the transversal and between lines p and q.
5. Angles 1 and 7 are angles because they
are on the transversal and outside lines p and q.
6. Angles 3 and 6 are angles because they
are on opposite sides of the transversal and between lines p
and q.
7. Angles 1 and 8 are angles because they are on opposite
sides of the transversal and outside lines p and q.
8. Angles 3 and 5 are angles because they are on the same
side of the transversal and between lines p and q.
9. Angles 2 and 8 are angles because they are on the same
side of the transversal and outside lines p and q.
10. Angles 3 and 7 are angles because they are on the same
side of the transversal in corresponding positions.
q
p
657 8
21
3 4
334 Chapter 2 ● Skills Practice
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Problem SetEach sketch shows two lines. Explain the relationship between the lines in each sketch.
1. p
q
The lines are coplanar and intersect at a single point.
2. p
q
3. p q
4. p
q
Identify the transversal in each diagram.
5. x
y
z
6. a
c
b
Line y
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Name _____________________________________________ Date ____________________
2
7.
g
h
f
8. k
l
m
Identify all pairs of vertical angles in each diagram.
9.
y
z
x1 2
3 4
5 6
7 8
10.
r
1 265
3 487
s
t
�1 and �4, �2 and �3, �5 and �8, �6 and �7
11.
1 562
3 7
84
m
n
p
12.
21
43
65
87
a b
c
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Identify all interior angles and all exterior angles in each diagram.
13.
t
r s
1 4
2 3
5 8
6 7
14.
1 243
5 687
x
y
z
Interior: �3, �4, �5, �6
Exterior: �1, �2, �7, �8
15.
1 265
3 487
a b
c
16.
24
31
6 875
kl
m
Identify all pairs of alternate interior angles and all pairs of alternate exterior angles in each diagram.
17. m
n
l
12
3
4
56
7
8
18. a
1 5
62
3 784
b
c
Alternate interior: �2 and �5, �3 and �8
Alternate exterior: �1 and �6, �4 and �7
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Name _____________________________________________ Date ____________________
2
19.
2 431
6 875
gh
j
20.
1 2
m
p
q
65
34
87
Identify all pairs of same side interior angles and all pairs of same side exterior angles in each diagram.
21. g
h
f
142
36
7 85
22. r s
t
1 2
78
3 456
Same side interior: �3 and �5, �4 and �6
Same side exterior: �1 and �7, �2 and �8
23. c
d
e
13
42
5 786
24. j
k
l
1 265
3 487
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Identify all pairs of corresponding angles in each diagram.
25. d
e
f
13
24
57
68
26. k
l
m
1 265
3 487
�1 and �5, �2 and �6, �3 and �7, �4 and �8
27. b
a
c
3 421
7 8
65
28. j
ik
5 621
3 487
Use a protractor to measure all eight angles in each diagram. Label the measure of each angle.
29. 60°
60°
60°
60°
120°120°
120°120°
30.
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31. 32.
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Skills Practice Skills Practice for Lesson 2.2
Name _____________________________________________ Date ____________________
Making ConjecturesConjectures about Angles Formed by Parallel Lines Cut by a Transversal
VocabularyExplain how each set of terms are related.
1. Corresponding Angle Postulate and corresponding angles
2. Alternate interior angles and alternate exterior angles
3. Same side interior angles and same side exterior angles
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Problem SetWrite congruence statements for the pairs of corresponding angles in each figure.
1. j
kl
12
34
56
78
2.
1 2
a
b
c
34
5 678
�1 � �5, �2 � �6, �3 � �7, �4 � �8
3. g
h
i1 265
3 487
4. mn
p1
3
24
57
6
8
Name _____________________________________________ Date ____________________
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Explain how you know that each statement is true.
5. �3 � �6 6. m�1 � m�4 � 180°
f
g
h
3
42
1
7
86
5
5
62
1
s
q
r
7
84
3
Alternate interior angles are congruent.
7. �1 � �5 8. �4 � �6
p q
n1 2
783 4
56
a c
b
15
73
26
84
9. m�4 � m�5 � 180° 10. �5 � �8
x
y
z
1 234
5 6
78
ef
g
1 2
65
7 8
43
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11. �6 � �8 12. �6 � �7
k l
m12
78
34
56
a
b
c
28
71
6
43
5
Use the given information to determine the measures of all unknown angles in each figure.
13. m�4 � 65° 14. m�8 � 155°
p
l
m34
21
68
75
a
b
c
1 234
5 6
78
m�1 � 65°, m�2 � 115°, m�3 � 115°, m�5 � 65°, m�6 � 115°, m�7 � 115°,
m�8 � 65°
Name _____________________________________________ Date ____________________
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15. m�6 � 89° 16. m�3 � 45°
x y
z
1 2 3 4
8765
l
m
n
13
57
86
42
17. m�7 � 30° 18. m�5 � 80°
k s
t13
2
7
68
5
4
w y
x
1 2
34
56
78
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19. m�4 � 95° 20. m�7 � 140°
d
e f
12
34
56
78
hi
j
21
5 6
43
7 8
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Skills Practice Skills Practice for Lesson 2.3
Name _____________________________________________ Date ____________________
What’s Your Proof?Alternate Interior Angle Theorem, Alternate Exterior Angle Theorem, Same-Side Interior Angle Theorem, and Same-Side Exterior Angle Theorem
VocabularyDefine each theorem in your own words.
1. Alternate Interior Angle Theorem
2. Alternate Exterior Angle Theorem
3. Same Side Interior Angle Theorem
4. Same Side Exterior Angle Theorem
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Problem SetDraw and label a diagram to illustrate each theorem.
1. Same Side Interior Angle Theorem
�1 and �3 are supplementary or �2 and �4 are supplementary
a
b
c
1 2
3 4
2. Alternate Exterior Angle Theorem
3. Alternate Interior Angle Theorem
Name _____________________________________________ Date ____________________
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4. Same Side Exterior Angle Theorem
Use the diagram to write the “given” and “prove” statements for each theorem.
5. If two parallel lines are cut by a transversal, then the
r
c
n
1 2 3 4
57 8
6
exterior angles on the same side of the transversal
are supplementary.
Given: r � c, n is a transversal
Prove: �1 and �7 are supplementary or �2 and �8 are supplementary
6. If two parallel lines are cut by a transversal,
t k
b
1 23 4
578
6
then the alternate exterior angles are congruent.
Given:
Prove:
7. If two parallel lines are cut by a transversal,
a zd
1 265 3 4
87
then the alternate interior angles are congruent.
Given:
Prove:
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8. If two parallel lines are cut by a transversal,
p
1 234 5 6
78
sw
then the interior angles on the same side
of the transversal are supplementary.
Given:
Prove:
Prove each statement using the indicated type of proof.
9. Use a paragraph proof to prove the a
b
c
12
34
56
78
Alternate Interior Angles Theorem. In
your proof, use the following
information and refer to the diagram.
Given: a � b, c is a transversal
Prove: �2 � �8
You are given that lines a and b are parallel and line c is a transversal, as shown in the diagram. Angles 2 and 6 are corresponding angles by definition, and corresponding angles are congruent by the Corresponding Angles Postulate. So, �2 � �6. Angles 6 and 8 are vertical angles by definition, and vertical angles are congruent by the Vertical Angles Congruence Theorem. So, �6 � �8. Since �2 � �6 and �2 � �8, by the Transitive Property, �2 � �8.
10. Use a two-column proof to prove the Alternate 1 2
5 6 3 47 8
r s
t
Exterior Angles Theorem. In your proof, use the
following information and refer to the diagram.
Given: r � s, t is a transversal
Prove: �4 � �5
Name _____________________________________________ Date ____________________
Chapter 2 ● Skills Practice 351
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11. Use a flow chart proof to prove the Same Side
12
56
3
478
x
y
z
Interior Angles Theorem. In your proof, use the
following information and refer to the diagram.
Given: x � y, z is a transversal
Prove: �6 and �7 are supplementary
12. Use a two-column proof to prove the Same Side Exterior Angles Theorem. In your
proof, use the following information and refer to the diagram.
Given: f � g, h is a transversal
Prove: �1 and �4 are supplementary
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Write the theorem that is illustrated by each statement and diagram.
13. �4 and �7 are supplementary
d
g
s
12 34
56 78
Same Side Exterior Angles Theorem
14. �2 � �6
71 2
85
3 4
6
q w
f
15. �1 � �8
k
12
43
56
87 n
t
16. �2 and �5 are supplementary
y1
v
p
243
5 687
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Skills Practice Skills Practice for Lesson 2.4
Name _____________________________________________ Date ____________________
A Reversed ConditionParallel Line Converse Theorems
VocabularyAnswer the following question.
1. What is the converse of a statement?
Problem Set
Write the converse of each postulate or theorem.
1. Corresponding Angle Postulate:
If a transversal intersects two parallel lines, then the corresponding angles formed
are congruent.
If corresponding angles formed by two lines and a transversal are congruent, then the two lines are parallel.
2. Alternate Interior Angle Theorem:
If a transversal intersects two parallel lines, then the alternate interior angles formed
are congruent.
3. Alternate Exterior Angle Theorem:
If a transversal intersects two parallel lines, then the alternate exterior angles formed
are congruent.
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4. Same-side Interior Angle Theorem:
If a transversal intersects two parallel lines, then the interior angles on the same side
of the transversal formed are supplementary.
5. Same-side Exterior Angle Theorem:
If a transversal intersects two parallel lines, then the exterior angles on the same
side of the transversal formed are supplementary.
Write the converse of each statement.
6. If a triangle has three congruent sides, then the triangle is an equilateral triangle.
Converse: If a triangle is an equilateral triangle, then the triangle has three congruent sides.
7. If a figure has four sides, then it is a quadrilateral.
8. If a figure is a rectangle, then it has four sides.
9. If two angles are vertical angles, then they are congruent.
Name _____________________________________________ Date ____________________
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10. If two angles in a triangle are congruent, then the triangle is isosceles.
11. If two intersecting lines form a right angle, then the lines are perpendicular.
Draw and label a diagram to illustrate each theorem.
12. Same-side Interior Angle Converse Theorem
Given: �1 and �3 are supplementary or �2 and �4 are supplementary
1 2
3 4
a
b
c
Conclusion: Lines a and b are parallel.
13. Alternate Exterior Angle Converse Theorem
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14. Alternate Interior Angle Converse Theorem
15. Same-Side Exterior Angle Converse Theorem
Use the diagram to write the “given” and “prove” statements for each theorem.
16. If two lines, cut by a transversal, form same-side
w
k
s
1 24 3
58 7
6
exterior angles that are supplementary, then
the lines are parallel.
Given: s is a transversal; �1 and �8 are supplementary or �2 and �7 are supplementary
Prove: w � k
Name _____________________________________________ Date ____________________
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17. If two lines, cut by a transversal, form alternate exterior lm
n
1 2
653 4
87
angles that are congruent, then the lines are parallel.
Given:
Prove:
18. If two lines, cut by a transversal, form alternate interior a
b
c
12
43
56
87
angles that are congruent, then the lines are parallel.
Given:
Prove:
19. If two lines, cut by a transversal, form same-side interior x
y
z
8
7
4
31
2
5
6
angles that are supplementary, then the lines are parallel.
Given:
Prove:
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Prove each statement using the indicated type of proof.
20. Use a paragraph proof to prove the Alternate Exterior Angles Converse Theorem.
In your proof, use the following information and refer to the diagram.
Given: �4 � �5, j is a transversal
Prove: p � x
p
x
j
12
65
348
7
You are given that �4 � �5 and line j is a transversal, as shown in the diagram. Angles 5 and 2 are vertical angles by definition, and vertical angles are congruent by the Vertical Angles Congruence Theorem. So, �5 � �2. Since �4 � �5 and �5 � �2, by the Transitive Property, �4 � �2. Angles 4 and 2 are corresponding angles by definition, and they are also congruent, so by the Corresponding Angles Converse Postulate, p � x.
21. Use a two column proof to prove the Alternate Interior Angles Converse Theorem.
In your proof, use the following information and refer to the diagram.
Given: �2 � �7, k is a transversal
Prove: m � n
n
m
k
12
65
34
87
Name _____________________________________________ Date ____________________
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22. Use a two column proof to prove the Same Side Exterior Angles Converse
Theorem. In your proof, use the following information and refer to the diagram.
Given: �1 and �4 are supplementary, u is a transversal
Prove: t � v
t
v
u
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23. Use a flow chart to prove the Same Side Interior Angles Converse Theorem. In your
proof, use the following information and refer to the diagram.
Given: �6 and �7 are supplementary, e is a transversal
Prove: f � g
g
f
e
1265
34
87
Write the theorem that is illustrated by each statement and diagram.
24. Lines r and s are parallel.
t
r
s
40°
140°
Same-side Interior Angles Converse Theorem
Name _____________________________________________ Date ____________________
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25. Lines g and h are parallel.
f
g
h
25°
25°
26. Lines b and c are parallel.
a
b
c120°
120°
27. Lines x and z are parallel.
x
z
y
150°
30°
362 Chapter 2 ● Skills Practice
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Chapter 2 ● Skills Practice 363
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Skills Practice Skills Practice for Lesson 2.5
Name _____________________________________________ Date ____________________
Many SidesNaming Geometric Figures
VocabularyMatch each term to its corresponding definition.
1. concave a. the simplest closed three-sided figure
2. consecutive angles b. closed geometric figure with four sides
3. consecutive sides c. sides of a figure that share a common angle
4. convex d. two angles in a figure that share a common side
5. decagon e. two angles of a quadrilateral that do not share a
common side
6. diagonal
f. a line segment of a closed figure whose endpoints
are two vertices that do not share a common side
7. nonagon
g. closed geometric figure where line segments
connecting any two points in the interior of the figure
are contained completely in the interior of the figure
8. hexagon
h. a polygon with all sides and all angles congruent
9. irregular polygon
i. five-sided polygon
10. octagon
j. two sides of a quadrilateral that do not share
an angle
11. opposite angles
k. six-sided polygon
12. opposite sides
l. a geometric figure that is not convex
13. pentagon
m. ten-sided polygon
14. polygon
n. an angle greater than 180° but less than 360°
15. quadrilateral
o. a closed figure that is formed by connecting three or
more line segments at their endpoints.
16. reflex angle
p. a polygon that is not regular
17. regular polygon
q. eight-sided polygon
18. triangle
r. nine-sided polygon
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Problem SetClassify each polygon shown.
1. 2.
triangle
3. 4.
5. 6.
7. 8.
Name _____________________________________________ Date ____________________
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Draw an example of each polygon. Label the vertices.
9. triangle ABC 10. hexagon HIJKLM
A C
B
11. quadrilateral XYZA 12. pentagon QRSTU
Construct an example of each polygon described using the given sides. Label the sides on the construction.
13. triangle 14. triangle
side aside b
side c
side a side b side c
side b
side a
side c
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15. quadrilateral 16. pentagon
side a side b
side d
side c
side e
side b
side a
side cside d
Name two pairs of consecutive angles and two pairs of consecutive sides for each quadrilateral.
17. G R
K W
18. M
O
N
P
Consecutive angles:
�K and �W, �W and �R, �R and �G, �G and �K
Consecutive sides:
____
KW and ____
WR , ____
WR and ____
RG ,
____ RG and
____ GK ,
____ GK and
____ KW
Name _____________________________________________ Date ____________________
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19. G
J H
P
20. FY
U Z
Name two pairs of opposite angles and two pairs of opposite sides for each quadrilateral.
21. D T
F
X
22. B
R
Q A
Opposite angles:
�X and �T, �D and �F
Opposite sides:
___
XD and ___
FT , ___
DT and ___
XF
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23.
L
E M
W
24.
T
X
Z K
Draw one diagonal for each polygon and name the diagonal.
25. B Y
LH
26.
M
R L
G
A diagonal is ___
BL ( or ___
HY )
27. T
P
R
L
28.
K
B
X P
Name _____________________________________________ Date ____________________
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Classify each polygon as concave or convex and regular or irregular.
29. 30.
concave and irregular
31. 32.
370 Chapter 2 ● Skills Practice
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Skills Practice Skills Practice for Lesson 2.6
Name _____________________________________________ Date ____________________
Quads and TrisClassifying Triangles and Quadrilaterals
VocabularyDraw an example of each term.
1. equilateral triangle 2. equiangular triangle
3. isosceles triangle 4. scalene triangle
5. acute triangle 6. right triangle
7. obtuse triangle 8. square
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9. rectangle 10. parallelogram
11. rhombus 12. kite
13. trapezoid
14. Provide a counterexample of the statement below.
All right triangles are scalene.
Name _____________________________________________ Date ____________________
Chapter 2 ● Skills Practice 373
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Problem SetClassify each triangle by its sides.
1. 2.
isosceles triangle
3. 4.
5. 6.
Classify each triangle by its angles.
7. 8.
acute triangle
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9. 10.
11. 12.
Construct each triangle described.
13. Construct an equilateral triangle using the given side.
14. Construct an equilateral triangle using the given side.
Name _____________________________________________ Date ____________________
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15. Construct an isosceles triangle using one of the given congruent sides.
16. Construct an isosceles triangle using one of the given congruent sides.
Draw an example of each triangle described.
17. scalene right triangle 18. scalene obtuse triangle
19. equilateral equiangular triangle 20. isosceles right triangle
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Classify each quadrilateral.
21. 22.
trapezoid
23. 24.
25. 26.
Construct each quadrilateral described.
27. Construct a square using the given side.
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28. Construct a rectangle using the given non-congruent sides.
29. Construct a rhombus using the given side.
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30. Construct a parallelogram using the given non-congruent sides.
31. Construct a kite using the given non-congruent sides.
32. Construct a trapezoid using the given non-congruent sides.
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Determine whether each statement is always true, sometimes true, or never true. Explain your answer.
33. All equilateral triangles are isosceles triangles.
Always true. An equilateral triangle is a triangle whose sides are congruent. An isosceles triangle is a triangle that has at least two congruent sides. An equilateral triangle has at least two congruent sides (it has three), so all equilateral triangles are also isosceles triangles.
34. All rectangles are squares.
35. All right triangles are acute triangles.
36. All rhombi are parallelograms.
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