Download Skills Practice Chapter 2

Skills Practice
Skills Practice for Lesson 2.1
Name _____________________________________________
Date ____________________
Transversals and Lines
Angles Formed by Transversals of Parallel and
Non-Parallel Lines
Vocabulary
Write 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
2
same side exterior
1. Coplanar lines that never intersect are called
2. Non-coplanar lines are called
lines.
lines.
3. A line that intersects two or more lines at distinct points is called
a(n)
.
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4. Angles 3 and 4 are
angles because they
are on the transversal and between lines p and q.
3
5. Angles 1 and 7 are
angles because they
are on the transversal and outside lines p and q.
p
1 2
4
q
5 6
7 8
6. Angles 3 and 6 are
angles because they
are on opposite sides of the transversal and between lines p
and q.
angles because they are on opposite
7. Angles 1 and 8 are
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.
angles because they are on the same
9. Angles 2 and 8 are
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.
Chapter 2 ● Skills Practice
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Problem Set
Each 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
2
3.
p
q
4.
p
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q
Identify the transversal in each diagram.
5.
6.
x
b
a
y
c
z
Line y
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Name _____________________________________________
7.
Date ____________________
8.
k
f
g
l
h
m
Identify all pairs of vertical angles in each diagram.
9.
10.
x
1
3
5
2
2
y
4
r
6
z
8
7
s
1 2
5 6
3 4
7 8
t
1 and 4, 2 and 3,
5 and 8, 6 and 7
11.
12.
m
n
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1 5
2 6
3
b
a
2
1
c
4
3
5
6
7
4 8
8
7
p
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Identify all interior angles and all exterior angles in each diagram.
13.
r
14.
s
1 2
3 4
t
7
6
3
2
y
8
5
4
1
x
5 6
7 8
z
Interior: 3, 4, 5, 6
Exterior: 1, 2, 7, 8
2
15.
a
16.
b
l
k
1 2
5 6
2
c
4
1
3
6 8
5 7
m
Identify all pairs of alternate interior angles and all pairs of alternate exterior
angles in each diagram.
17.
l
1 3
4 2
18.
m
5 7
8 6
n
a
1 5
2 6
c
3
4
Alternate interior:
2 and 5, 3 and 8
Alternate exterior:
1 and 6, 4 and 7
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Chapter 2 ● Skills Practice
b
7
8
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3 4
7 8
Name _____________________________________________
19.
g
20.
h
2 4
1 3
6 8
5 7
Date ____________________
m
1 2
5 6
j
3
7
p
4
8
q
2
Identify all pairs of same side interior angles and all pairs of same side
exterior angles in each diagram.
21.
22.
f
r
s
g
1 2
3 4
5 6
7 8
h
1 2
8 7
t
3 4
6 5
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Same side interior:
3 and 5, 4 and 6
Same side exterior:
1 and 7, 2 and 8
23.
24.
c
1
2
j
1 2
5 6
3
k
4
5
7
6 8
3 4
7 8
d
l
e
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Identify all pairs of corresponding angles in each diagram.
25.
1
3
4
8
5
7
26.
d
2
l
k
e
6
1 2
5 6
f
3 4
7 8
m
1 and 5, 2 and 6,
3 and 7, 4 and 8
27.
b
28.
c
j
i
a
3 4
1 2
5 6
1 2
7 8
5 6
k
3 4
7 8
Use a protractor to measure all eight angles in each diagram. Label the
measure of each angle.
29.
30.
60° 120°
120° 60°
60° 120°
120° 60°
338
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Name _____________________________________________
31.
Date ____________________
32.
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2
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2
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Chapter 2 ● Skills Practice
Skills Practice
Name _____________________________________________
Skills Practice for Lesson 2.2
Date ____________________
Making Conjectures
Conjectures about Angles Formed by Parallel Lines
Cut by a Transversal
Vocabulary
Explain how each set of terms are related.
1. Corresponding Angle Postulate and corresponding angles
2
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2. Alternate interior angles and alternate exterior angles
3. Same side interior angles and same side exterior angles
Chapter 2 ● Skills Practice
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Problem Set
Write congruence statements for the pairs of corresponding angles in
each figure.
1.
k
j
4
1
3
l
2.
a
8
5
7
b
1 2
4 3
2
6
5 6
8 7
2
c
1 ⬵ 5, 2 ⬵ 6,
3 ⬵ 7, 4 ⬵ 8
3.
g
4.
h
m
n
3
1
1
2
5 6
3
4
7 8
i
2
p
4
7
6
5
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8
342
Chapter 2 ● Skills Practice
Name _____________________________________________
Date ____________________
Explain how you know that each statement is true.
5. ⬔3 ⬵ ⬔6
6. m⬔1 ⫹ m⬔4 ⫽ 180°
q
g
f
1 5
2 6
h
7
5
6 8
3
1
2 4
r
3 7
4 8
s
Alternate interior angles
are congruent.
7. ⬔1 ⬵ ⬔5
2
8. ⬔4 ⬵ ⬔6
p
q
c
a
1
3
1 2
8 7
3 4
6 5
5
7
n
b
2
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4
9. m⬔4 ⫹ m⬔5 ⫽ 180°
6
8
10. ⬔5 ⬵ ⬔8
e
z
f
x
1 2
4 3
y
5
8
1 2
5 6
g
6
7
7 8
3 4
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11. ⬔6 ⬵ ⬔8
12. ⬔6 ⬵ ⬔7
l
k
b
a
c
m
1
5
2
8
3
4
6
3
2
1 8
7
7
6
4
5
2
Use the given information to determine the measures of all unknown angles
in each figure.
13. m⬔4 ⫽ 65°
14. m⬔8 ⫽ 155°
l
a
p
m
4
2
6
5
b
2
4
3
8
7
6
5
8
m1 ⴝ 65°, m2 ⴝ 115°, m3 ⴝ 115°,
m5 ⴝ 65°, m6 ⴝ 115°, m7 ⴝ 115°,
m8 ⴝ 65°
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Chapter 2 ● Skills Practice
c
7
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1
1
Name _____________________________________________
15. m⬔6 ⫽ 89°
Date ____________________
16. m⬔3 ⫽ 45°
x
m
y
l
1 2
5 6
3
4
7 8
5
z
n
7
8
6
1
3
4
2
2
17. m⬔7 ⫽ 30°
18. m⬔5 ⫽ 80°
k
1
2
t
x
7
6
5
8
6
7
© 2010 Carnegie Learning, Inc.
8
y
1 2
4 3
3
4
5
w
s
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19. m⬔4 ⫽ 95°
20. m⬔7 ⫽ 140°
e
d
f
3
6
5
2
1
8
h
i
4
2
1
5
6
j
7
4
3
7
8
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2
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Chapter 2 ● Skills Practice
Skills Practice
Name _____________________________________________
Skills Practice for Lesson 2.3
Date ____________________
What’s Your Proof?
Alternate Interior Angle Theorem, Alternate Exterior
Angle Theorem, Same-Side Interior Angle Theorem, and
Same-Side Exterior Angle Theorem
Vocabulary
Define each theorem in your own words.
1. Alternate Interior Angle Theorem
2
2. Alternate Exterior Angle Theorem
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3. Same Side Interior Angle Theorem
4. Same Side Exterior Angle Theorem
Chapter 2 ● Skills Practice
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Problem Set
Draw 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
c
1 2
a
3 4
b
2
3. Alternate Interior Angle Theorem
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Chapter 2 ● Skills Practice
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2. Alternate Exterior Angle Theorem
Name _____________________________________________
Date ____________________
4. Same Side Exterior Angle Theorem
2
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
exterior angles on the same side of the transversal
are supplementary.
n
3
Given: r 储 c, n is a transversal
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Prove: 1 and 7 are supplementary or 2 and 8
are supplementary
7
5
8
6. If two parallel lines are cut by a transversal,
then the alternate exterior angles are congruent.
Given:
1
2
8 7
1
4
2
6
3
r
c
4
6
b
5
Prove:
t
7. If two parallel lines are cut by a transversal,
then the alternate interior angles are congruent.
1 2
5 6
3 4
7 8
Given:
Prove:
a
k
d
z
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8. If two parallel lines are cut by a transversal,
then the interior angles on the same side
of the transversal are supplementary.
1 2
4 3
5 6
8 7
p
Given:
w
Prove:
s
Prove each statement using the indicated type of proof.
1
4
a
2
3
b
5
8
6
7
c
Given: a 储 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
Exterior Angles Theorem. In your proof, use the
following information and refer to the diagram.
Given: r 储 s, t is a transversal
Prove: ⬔4 ⬵ ⬔5
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Chapter 2 ● Skills Practice
1
2
5
3
6
r
4
7 8
t
s
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2
9. Use a paragraph proof to prove the
Alternate Interior Angles Theorem. In
your proof, use the following
information and refer to the diagram.
Name _____________________________________________
11. Use a flow chart proof to prove the Same Side
Interior Angles Theorem. In your proof, use the
following information and refer to the diagram.
Given: x 储 y, z is a transversal
Prove: ⬔6 and ⬔7 are supplementary
Date ____________________
z
x
15
26
y
3
7
4 8
2
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 储 g, h is a transversal
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Prove: ⬔1 and ⬔4 are supplementary
Chapter 2 ● Skills Practice
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Write the theorem that is illustrated by each statement and diagram.
13. ⬔4 and ⬔7 are supplementary
d
6
5 8
2
1 4
7
s
3
g
Same Side Exterior Angles Theorem
14. ⬔2 ⬵ ⬔6
w
q
1 2
8 7
3 4
6 5
f
15. ⬔1 ⬵ ⬔8
k
1
2
3
4
5
6
7
16. ⬔2 and ⬔5 are supplementary
y
1 2
3 4
5 6
78
p
v
352
Chapter 2 ● Skills Practice
t
8
n
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2
Skills Practice
Name _____________________________________________
Skills Practice for Lesson 2.4
Date ____________________
A Reversed Condition
Parallel Line Converse Theorems
Vocabulary
Answer the following question.
1. What is the converse of a statement?
2
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.
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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.
Chapter 2 ● Skills Practice
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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.
2
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.
8. If a figure is a rectangle, then it has four sides.
9. If two angles are vertical angles, then they are congruent.
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7. If a figure has four sides, then it is a quadrilateral.
Name _____________________________________________
Date ____________________
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.
2
12. Same-side Interior Angle Converse Theorem
Given: 1 and 3 are supplementary or 2 and 4 are supplementary
c
1 2
a
3 4
b
© 2010 Carnegie Learning, Inc.
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
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 储 k
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Chapter 2 ● Skills Practice
s
4
8
5
7
6
1 2
3
w
k
© 2010 Carnegie Learning, Inc.
2
Name _____________________________________________
Date ____________________
17. If two lines, cut by a transversal, form alternate exterior
angles that are congruent, then the lines are parallel.
l
m
Given:
1
2
5
3
6
4
7 8
n
Prove:
18. If two lines, cut by a transversal, form alternate interior
angles that are congruent, then the lines are parallel.
Given:
a
b
2
1
4
3
5
7
Prove:
19. If two lines, cut by a transversal, form same-side interior
angles that are supplementary, then the lines are parallel.
Given:
y
8
z
1 3
2 4
5 7
6
8
© 2010 Carnegie Learning, Inc.
Prove:
x
c
6
Chapter 2 ● Skills Practice
357
2
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 储 x
5
1
6
p
2
x
3
8
7
4
j
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 储 x.
2
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 储 n
1
6
n
2
m
7
358
Chapter 2 ● Skills Practice
3
8
4
k
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5
Name _____________________________________________
Date ____________________
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 储 v
5
1
6
t
2
v
7
3
8
4
u
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2
Chapter 2 ● Skills Practice
359
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 储 g
5
1
6
g
2
f
3
8
7
4
e
Write the theorem that is illustrated by each statement and diagram.
24. Lines r and s are parallel.
t
40°
s
140°
r
Same-side Interior Angles Converse Theorem
360
Chapter 2 ● Skills Practice
© 2010 Carnegie Learning, Inc.
2
Name _____________________________________________
Date ____________________
25. Lines g and h are parallel.
f
25°
g
25°
h
26. Lines b and c are parallel.
2
b
a
c
120°
120°
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27. Lines x and z are parallel.
z
x
30°
150°
y
Chapter 2 ● Skills Practice
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2
362
Chapter 2 ● Skills Practice
Skills Practice
Skills Practice for Lesson 2.5
Name _____________________________________________
Date ____________________
Many Sides
Naming Geometric Figures
Vocabulary
Match 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
7. nonagon
8. hexagon
f. a line segment of a closed figure whose endpoints
are two vertices that do not share a common side
10. octagon
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
11. opposite angles
h. a polygon with all sides and all angles congruent
9. irregular polygon
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2
12. opposite sides
i. five-sided polygon
13. pentagon
j. two sides of a quadrilateral that do not share
an angle
14. polygon
k. six-sided polygon
15. quadrilateral
l. a geometric figure that is not convex
16. reflex angle
m. ten-sided polygon
17. regular polygon
n. an angle greater than 180° but less than 360°
18. triangle
o. a closed figure that is formed by connecting three or
more line segments at their endpoints.
p. a polygon that is not regular
q. eight-sided polygon
r. nine-sided polygon
Chapter 2 ● Skills Practice
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Problem Set
Classify each polygon shown.
1.
2.
triangle
3.
4.
5.
6.
7.
8.
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Chapter 2 ● Skills Practice
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2
Name _____________________________________________
Date ____________________
Draw an example of each polygon. Label the vertices.
9. triangle ABC
10. hexagon HIJKLM
B
A
C
11. quadrilateral XYZA
12. pentagon QRSTU
2
Construct an example of each polygon described using the given sides.
Label the sides on the construction.
13. triangle
14. triangle
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side c
side a
side b
side b
side a
side b
side c
side c
side a
Chapter 2 ● Skills Practice
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15. quadrilateral
16. pentagon
side a
side a
side d
side c
side b
side e
side b
side d
side c
2
Name two pairs of consecutive angles and two pairs of consecutive sides
for each quadrilateral.
R
G
K
W
K and W, W and R,
R and G, G and K
____
____ ____
____
KW
WR ,____
WR and____
RG ,
____ and ____
RG and GK , GK and KW
366
Chapter 2 ● Skills Practice
M
O
Consecutive angles:
Consecutive sides:
18.
N
P
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17.
Name _____________________________________________
Date ____________________
G
19.
20.
H
F
Y
J
U
Z
P
2
Name two pairs of opposite angles and two pairs of opposite sides for
each quadrilateral.
21.
D
T
22.
B
R
X
F
© 2010 Carnegie Learning, Inc.
Q
A
Opposite angles:
X and T, D and F
Opposite sides:
___
___ ___
___
XD and FT , DT and XF
Chapter 2 ● Skills Practice
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23.
E
24.
M
X
T
L
W
K
Z
2
Draw one diagonal for each polygon and name the diagonal.
25.
B
Y
26.
L
R
G
H
___
L
M
___
A diagonal is BL ( or HY )
27.
T
28.
B
L
P
R
X
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P
© 2010 Carnegie Learning, Inc.
K
Name _____________________________________________
Date ____________________
Classify each polygon as concave or convex and regular or irregular.
29.
30.
concave and irregular
31.
32.
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2
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2
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Skills Practice
Skills Practice for Lesson 2.6
Name _____________________________________________
Date ____________________
Quads and Tris
Classifying Triangles and Quadrilaterals
Vocabulary
Draw an example of each term.
1. equilateral triangle
2. equiangular triangle
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2
3. isosceles triangle
4. scalene triangle
5. acute triangle
6. right triangle
7. obtuse triangle
8. square
Chapter 2 ● Skills Practice
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9. rectangle
11. rhombus
10. parallelogram
12. kite
2
13. trapezoid
All right triangles are scalene.
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14. Provide a counterexample of the statement below.
Name _____________________________________________
Date ____________________
Problem Set
Classify each triangle by its sides.
1.
2.
isosceles triangle
3.
4.
2
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5.
6.
Classify each triangle by its angles.
7.
8.
acute triangle
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9.
10.
11.
12.
2
Construct each triangle described.
14. Construct an equilateral triangle using the given side.
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Chapter 2 ● Skills Practice
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13. Construct an equilateral triangle using the given side.
Name _____________________________________________
Date ____________________
15. Construct an isosceles triangle using one of the given congruent sides.
16. Construct an isosceles triangle using one of the given congruent sides.
2
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Draw an example of each triangle described.
17. scalene right triangle
18. scalene obtuse triangle
19. equilateral equiangular triangle
20. isosceles right triangle
Chapter 2 ● Skills Practice
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Classify each quadrilateral.
21.
22.
2
23.
24.
25.
26.
Construct each quadrilateral described.
27. Construct a square using the given side.
376
Chapter 2 ● Skills Practice
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trapezoid
Name _____________________________________________
Date ____________________
28. Construct a rectangle using the given non-congruent sides.
2
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29. Construct a rhombus using the given side.
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30. Construct a parallelogram using the given non-congruent sides.
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32. Construct a trapezoid using the given non-congruent sides.
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© 2010 Carnegie Learning, Inc.
31. Construct a kite using the given non-congruent sides.
Name _____________________________________________
Date ____________________
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.
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35. All right triangles are acute triangles.
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36. All rhombi are parallelograms.
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