Transversals and Lines Angles Formed by …...2011/11/07 · Conjectures about Angles Formed by...

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.


are on the transversal and outside lines p and q.


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.


side of the transversal and outside lines p and q.


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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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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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

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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°

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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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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

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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:


a zd

1 265 3 4

87

then the alternate interior angles are congruent.

Given:

Prove:

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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

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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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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.

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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

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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

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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

1265

348

7

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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

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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°


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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.

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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

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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

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Classify each polygon as concave or convex and regular or irregular.

29. 30.

concave and irregular

31. 32.


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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.

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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.

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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.

Transversals and Lines Angles Formed by …...2011/11/07 · Conjectures about Angles Formed by...

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Transcript of Transversals and Lines Angles Formed by …...2011/11/07 · Conjectures about Angles Formed by...