Skills Practice - Mater Academy Charter Middle / High School

Chapter 6 ● Skills Practice 179

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Skills Practice Skills Practice for Lesson 6.1

Name _____________________________________________ Date__________________________

Visiting Washington, D.C.Transversals and Parallel Lines

Vocabulary Write the term from the box that best completes each statement.

plane transversal alternate exterior angles

coplanar interior angles same-side interior angles

parallel lines exterior angles same-side exterior angles

skew lines alternate interior angles corresponding angles

1. A(n) is a line that intersects two or more other lines in the

same plane at different points.

2. When two lines are cut by a transversal, are formed, which

are two angles that lie between the two lines and on opposite sides of the transversal.


are two angles that lie outside the two lines on the same side of the transversal.

4. are two lines in the same plane that do not intersect.

5. of two similar or congruent figures are pairs of angles that are

in the same relative position in both figures.


are any angles that lie outside the two lines on either side of the transversal.

7. are two lines that are not coplanar and do not intersect.

8. A(n) is a flat surface that extends without end in two

dimensions.


are any angles that lie inside the two lines on either side of the transversal.


are two angles that lie outside the two lines and on opposite sides of the transversal.

11. lines are lines that lie in the same plane.


are two angles that lie between the two lines on the same side of the transversal.

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180 Chapter 6 ● Skills Practice

6

Problem Set Determine whether the given lines appear to be parallel or whether they intersect.

13. lines a and b 14. lines a and c

a

b

ac

15. lines b and c 16. lines b and d

b c

b

d

Identify the indicated angles in each figure.

17. All the pairs of alternate interior angles 18. All the pairs of alternate exterior angles

m

1 2

3

5

7 8

6

4n

o

r

q

s

1

3

5

7 8

6

4

2


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Name _____________________________________________ Date _________________________

19. All the pairs of same-side interior angles 20. All the pairs of same-side exterior angles

r

q

s

1

3

5

7 8

6

4

2

w x

13 4

2 5

7 8

6

21. All the pairs of corresponding angles 22. All the interior angles

u

t

v

13 4

2 57 8

6

y z

t1 2 5

7 8

6

3 4

23. All the exterior angles

21 4

35

6

78

a

b c

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Determine the measure of each missing angle.

24. Given m�1 � 75º, what is m�3?

1

3

5

7 8

6

4

2

a

t

b


13 4

2 57 8

6t

c d


13 4

2 57 8

6t

e f


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Name _____________________________________________ Date _________________________


1 2 5

7 86

3 4t

g h

Given the measure of one angle, determine the measure of the other angles.

28. m�1 � 102º. Lines a and b are parallel.

13

57 8

6

42 a

b

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29. m�5 � 73º. Lines c and d are parallel.

1 23

57 8

6

4c

t

d

30. m�3 � 67º. Lines e and f are parallel.

13 4

2 57 8

6

e f

t


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Name _____________________________________________ Date _________________________

31. m�7 � 115º. Lines g and h are parallel.

1 2 5 67 8

3 4t

g h

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Name _____________________________________________ Date _________________________

Going Up?Introduction to Proofs

Vocabulary Define each term in your own words.

1. congruent angles

2. if-then form

3. hypothesis

4. proof

5. two-column proof

6. conclusion

7. conditional statement

8. postulate

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Problem Set Identify the hypothesis and the conclusion of each if-then statement.

9. If (x1, y

1) and (x

2, y

2) are two points in the coordinate plane, then the midpoint of the line

segment that joins these two points is given by ( x1 � x

2 _______ 2 ,

y1 � y

2 _______ 2 ) .

10. If (x1, y

1) and (x

2, y

2) are two points in the coordinate plane, then the distance, d, between

the two points is given by d � √___________________

(x2 � x

1)2 � (y

2 � y

1)2 .

11. If the angles of a triangle measure 30º, 60º, and 90º, then the length of the hypotenuse

is two times the length of the shorter leg, and the length of the longer leg is √__

3 times the

length of the shorter leg.

12. If the angles of a triangle measure 45º, 45º, and 90º, then the length of the hypotenuse is

√__

2 times the length of a leg.

13. If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse,

then a2 � b2 � c2.

14. If two lines m and n are in the same plane, then lines m and n are coplanar.


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Name _____________________________________________ Date _________________________

15. If two lines m and n are coplanar and do not intersect, then lines m and n are parallel.

16. If two lines m and n are not coplanar and do not intersect, then lines m and n are skew.

Complete each two-column proof.

17. If �1 and �2 are supplementary angles and �2 and �3 are supplementary angles, then

m�1 � m�3.

1

2 3

Statement Reason

1. �1 and �2 are supplementary angles. 1. Given

2. �2 and �3 are supplementary angles. 2.

3. m�1 � m�2 � 180º 3.

4. m�2 � m�3 � 180º 4. Definition of supplementary angles

5. 180º � m�2 � m�3 5. Property of Equality

6. m�1 � m�2 � m�2 � m�3 6. Property of Equality

7. m�1 � m�3 7. Subtraction Property of Equality

8. �1 � �3 8. Definition of

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18. If �4 and �5 are complementary angles and �5 and �6 are complementary angles, then

m�4 � m�6.

4 56

Statement Reason

1. �4 and �5 are complementary angles. 1. Given

2. �5 and �6 are complementary angles. 2.

3. m�4 � m�5 � 90º 3.

4. m�5 � m�6 � 90º 4. Definition of complementary angles

5. 90º � m�5 � m�6 5. Property of Equality

6. m�4 � m�5 � m�5 � m�6 6. Transitive Property of Equality

7. m�4 � m�6 7. Property of Equality

8. �4 � �6 8. Definition of

19. If �1 and �3 are vertical angles, then �1 � �3.

12

3

Statement Reason

1. �1 and �3 are . 1. Given

2. �1 and �2 form a . 2. Definition of linear pair


4. �1 and �2 are . 4. Linear Pair Postulate


6. 6. Congruent Supplements Theorem


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Name _____________________________________________ Date _________________________

20. If �4 and �6 are vertical angles, then m�4 � m�6.

45

6

Statement Reason

1. �4 and �6 are . 1. Given






21. If �1 is a right angle, then �2 is also a right angle.

1

3

2

4

Statement Reason

1. �1 is a right angle. 1.

2. m�1 � 90º 2. Definition of

3. �1 and �2 form a linear pair. 3. Definition of

4. �1 and �2 are supplementary. 4. Postulate

5. m�1 � m�2 � 180º 5. Definition of

6. 90º � m�2 � 180º 6. Property of Equality

7. 90º � m�2 � 90º � 180º � 90º 7. Property of Equality

8. m�2 � 90º 8. Simplify.

9. �2 is a right angle. 9. Definition of

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22. If �5 is a right angle, then �8 is also a right angle.

5

7

6

8

Statement Reason

1. �5 is a right angle. 1.

2. m�5 � 90º 2. Definition of

3. �5 and �8 are vertical angles. 3. Definition of

4. �5 � �8 4. Theorem

5. m�5 � m�8 5. Definition of

6. 90º � m�8 6. Property of Equality

7. m�8 � 90º 7. Property of Equality

8. �8 is a right angle. 8. Definition of


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Name _____________________________________________ Date _________________________

Working with IronParallel Lines and Proofs

Vocabulary Classify each pair of angles as corresponding angles, same-side interior angles, same-side exterior angles, alternate interior angles, or alternate exterior angles.

1

a

b

c

3

57 8

6

42

1. �1 and �5 2. �3 and �6

3. �4 and �6 4. �2 and �7

Problem Set Given a line l and transversal t, use a compass and straightedge to construct a line parallel to l.

5. l

t 6.

l

t

7. l

t

8. l

t

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9. l

t

10. l

t

11. l

t

12. l

t

Complete each two-column proof.

13. If �1 � �2, then line l is parallel to line m.

lm

1

2 3t

Statement Reason

1. 1. Given

2. 2. Vertical Angles Congruence Theorem

3. 3. Transitive Property of Congruence

4. 4. Corresponding Angles Postulate

14. If �1 � �2, then line l is parallel to line m.

lm

1

2 3t

Statement Reason

1. 1. Given

2. 2. Vertical Angles Congruence Theorem


4. 4. Corresponding Angles Postulate


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Name _____________________________________________ Date _________________________

15. If �1 and �2 are supplementary angles, then line l is parallel to line m.

lm

123

t

Statement Reason



3. �1 � �3 3.

4. l || m 4.

16. If �1 and �2 are supplementary angles and �2 and �4 are congruent, then line l is parallel

to line n.

lm

1 23 4

n

t

Statement Reason

1. �1 and �2 are supplementary. 1.

2. m�1 � m�2 � 180° 2.

3. �3 and �4 are supplementary. 3.

4. m�3 � m�4 � 180° 4.

5. m�1 � m�2 � m�3 � m�4 5.

6. �2 � �4 6.

7. m�2 � m�4 7.

8. m�1 � m�4 � m�3 � m�4 8.

9. m�1 � m�3 9.

10. �1 � �3 10.

11. l || n 11.

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17. If �1 and �2 are congruent and �2 and �3 are supplementary angles, then line l is parallel

to line n.

lm

12 3 4

n

t

Statement Reason

1. 1. Given

2. 2. Given

3. 3. Linear Pair Postulate



6. 6. Alternate Exterior Angles Theorem


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Name _____________________________________________ Date ________________________

Parallel LinesConstructing Parallel Lines

Vocabulary Explain why the figure is a rhombus.

1.

Problem Set Use a compass, a straightedge, and the converse of the Alternate Interior Angle Theorem to construct parallel lines.

2. Construct a line parallel to line r.

s

r

3. Construct a line parallel to line a.

a

b

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4. Construct any pair of parallel lines f and h with transversal g.

5. Construct any pair of parallel lines x and z with transversal y. Make sure the lines are in a

different position than the lines in Question 4.

Use a compass, a straightedge, and the converse of the Same-Side Interior Angle Theorem to construct parallel lines.


r

s

7. Construct a line parallel to line l.

l

m


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Name _____________________________________________ Date _________________________

8. Construct any pair of parallel lines r and t with transversal s.

9. Construct any pair of parallel lines a and c with transversal b. Make sure the lines are in a


Use a compass, a straightedge, and the converse of the Alternate Exterior Angle Theorem to construct parallel lines.

10. Given two intersecting lines, use a compass and straightedge to construct a line parallel to

line r.

s

r

11. Given two intersecting lines, use a compass and straightedge to construct a line parallel to

line p.

q

p

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12. Construct any pair of parallel lines a and c with transversal b.

13. Construct any pair of parallel lines r and t with transversal s. Make sure the lines are in a


Use a compass, a straightedge, and the rhombus method to construct parallel lines.

14. Use the two circles to construct parallel lines.

A B

15. Use the two circles to construct parallel lines.

E F


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Name _____________________________________________ Date _________________________

16. Use the circle to construct parallel lines.

A

17. Use the circle to construct parallel lines.

E

18. Use a compass and straightedge to construct rhombus ABDC.

19. Use a compass and straightedge to construct rhombus EFHG.

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Construct parallel lines using patty paper.


A

r


A

r

22. Construct lines that are parallel to ___

AB and ___

AC .

P

B

A C

23. Construct lines that are parallel to ___

AB and ___

AC .

P B

C A


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Name _____________________________________________ Date _________________________

Parking Lot DesignParallel and Perpendicular Lines in the Coordinate Plane

Vocabulary Write the term from the box that best completes each statement.

negative reciprocals slope point-slope form vertical line

perpendicular lines y-intercept horizontal line

slope-intercept form reciprocals parallel lines

1. Two numbers are if their product is �1.

2. The of a nonvertical line is the ratio of the vertical change to the

horizontal change.

3. The of a linear equation that passes through the point (x1, y

1) and

has slope m is y � y1 � m(x � x

1).

4. Two lines in the same plane are if they intersect at right angles.

5. The is the point at which a line intersects the y-axis.

6. A(n) has an equation of the form y � a where a is any real number.

7. The of a linear equation is y � mx � b, where m is the slope of the

line and b is the y-intercept of the line.

8. Two non-zero numbers are if their product is 1.

9. A(n) has an equation of the form x � b where b is any real number.

10. Two lines in the same plane are if they do not intersect.

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Problem Set Determine whether the lines are parallel, perpendicular, or neither. Explain your answer.

11. line l: y � �2x � 4

line m: y � �2x � 8

12. line p: y � 3x � 5

line q: y � 1 __ 3 x � 5

13. line r: y � �5x � 12

line s: y � 1 __ 5 x � 6

14. line t: y � 3x � 1

line u: y � 3x � 7

15. line l: y � 6x � 2

line m: y � �6x � 2

16. line n: y � x � 8

line o: y � �x � 1


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Name _____________________________________________ Date _________________________

Determine whether the lines shown on each graph are parallel, perpendicular, or neither. Explain your answer.

17.

p

q

x108 9

6

7

8(0, 8)

(9, 2)

(2, 0)

9

10

4 5

4

5

6 70

3

1 2 3

2

1

y

(8, 9)

18.

sr

x108 9

6

7

8

(1, 10)

(6, 0)(3, 0)

9

10

4 5

4

5

6 70

3

1 2 3

2

1

y

(8, 10)

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19.

t

u

x108 9

6

7

8

(7, 9)

(6, 0)(1, 0)

9

10

4 5

4

5

6 70

3

1 2 3

2

1

y

(10, 8)

20.

l

m

x108 9

6

7

8

(0, 3)

(10, 6)

(2, 0)

9

10

4 5

4

5

6 70

3

1 2 3

2

1

y

(8, 9)


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Name _____________________________________________ Date _________________________

Determine an equation of the parallel line described. Write your answer in both point-slope form and slope-intercept form.

21. What is the equation of a line parallel to y � 4 __ 5 x � 2 that passes through (1, 2)?

22. What is the equation of a line parallel to y � �5x � 3 that passes through (3, 1)?

23. What is the equation of a line parallel to y � 7x � 8 that passes through (5, �2)?

24. What is the equation of a line parallel to y � � 1 __ 2

x � 6 that passes through (�4, 1)?

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Determine an equation of the perpendicular line described. Write your answer in both point-slope form and slope-intercept form.

25. What is the equation of a line perpendicular to y � 2x � 6 that passes through (5, 4)?

26. What is the equation of a line perpendicular to y � �3x � 4 that passes through (�1, 6)?

27. What is the equation of a line perpendicular to y � � 2 __ 5 x � 1 that passes through (2, �8)?

28. What is the equation of a line perpendicular to y � 3 __ 4 x � 12 that passes through (12, 3)?


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Name _____________________________________________ Date _________________________

Determine the equation of a vertical line that passes through the given point.

29. (�2, 1) 30. (3, 15)

31. (9, �7) 32. (�11, �8)

Determine the equation of a horizontal line that passes through the given point.

33. (4, 7) 34. (�6, 5)

35. (�8, �3) 36. (2, �9)

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Name _____________________________________________ Date _________________________

Building a HengeExploring Triangles in the Coordinate Plane

Vocabulary Draw an example of each key term.

1. inscribed triangle

2

4

6

–6

–8

–2

–4

–2–4 2 4–8 –6 x6 8

8

y

2. midsegments of an inscribed triangle

2

4

6

–6

–8

–2

–4

–2–4 2 4–8 –6 x6 8

8

y

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Problem Set Use the diagram to determine the midpoints of each side of the inscribed triangle. Simplify your answers, but do not evaluate any radicals.

3.

1

2

3

–3

–4

–1

–2

–1–2 1 2–4

N

–3 x3 4

4

y

M

L

4.

2

4

6

–6

–8

–2

–4

–2–4 2 4–8

R

–6 x6 8

8

y

P

Q


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Name _____________________________________________ Date _________________________

5.

3

6

9

–9

–12

–3

–6

–3–6 3 6–12

D

–9 x9 12

12

y

EF

6.

2

4

6

–6

–8

–2

–4

–2–4 2 4–8

G

–6 x6 8

8

y

H

I

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Given the midpoint of the hypotenuse of an inscribed triangle, calculate the distance from the midpoint to each vertex.

7. The midpoint of EF is point G.

1

2

3

–3

–4

–1

–2

–1

(0,0)

–2 1 2–4 –3 x3 4

4

y

D

E G

H I(–2, –2) (2, –2)

F

8. The midpoint of GH is point M.

2

4(–4, 4)

6

–6

–8

–2

–4

–2–4 2(0, 0)

M

N

G

4–8 –6 x6 8

8

y

H

L(–4, –4)

I


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Name _____________________________________________ Date _________________________

9. The midpoint of BC is point D.

1

2

3

–3

–4

–1

–2

–1 (0, 0)

(1, 2)

–2 1 2–4 –3 x3 4

4

y

S

R

A

B

E F(–4, 2 )

C

D

10. The midpoint of RS is point X.

2

4

(–4, 8)

6

–6

–8

–2

–4

–2–4 2(0, 0)

XZ

YQ

R

4–8 –6 x6 8

8

y

S

(–4, –2)

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Given each diagram, compare the measures described. Simplify your answers, but do not evaluate any radicals.

11. Triangle LMN has midpoints O(0, 0), P(�9, �3), and Q(0, �3). Compare the length of OP to

the length of LM.

2

4

6

–6

–8

–2

–4

–2–4 2

MN

L

PQ

O

4–8 –6 x6 8

8

y


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Name _____________________________________________ Date _________________________

12. Triangle LMN has midpoints P(0, 0), Q(�2, 2), and R(2, 2). Compare the length of QP to the

length of LN.

1

2

3

–3

–4

–1

–2

–1(0, 0)

–2 1 2–4 –3 x3 4

4

y

Q

P

M

N

L

(–2, 2) (2, 2)R

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13. Triangle ABC has midpoints D(0, 0), E(6, 6), and F(6, �6). Compare the slope of DF to the

slope of AC.

3

6

9

–9

–12

–3

–6

–3–6 3 6–12

F

C–9 x9 12

12

y

E

A

B

D

14. Triangle PQR has midpoints S(6, �12), T(6, 3), and U(0, 0). Compare the slope of UT to the

slope of QP.

3

6

9

–9

–12

–3

–6

–3–6 3 6–12

S

P

–9 x9 12

12

y

T

R

Q

U


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Name _____________________________________________ Date _________________________

Building a Roof TrussAngle and Line Segment Bisectors

Vocabulary Define each term in your own words.

1. angle bisector

2. perpendicular bisector

3. bisect a line segment

4. bisect an angle

5. line segment bisector

Problem Set Use a compass and a straightedge to bisect each angle.

6. 7.

8. 9.

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10. 11.

12. 13.

Given the measure of an angle, calculate the measure of each new angle formed when the angle is bisected.

14. m� A � 36º 15. m�B � 48º

16. m�C � 144º 17. m�D � 126º

18. m�E � 55º 19. m�F � 67º

20. m�G � 171º 21. m�H � 139º


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Name _____________________________________________ Date _________________________

Use a compass and straightedge to bisect each angle of a triangle.

22. 23.

24. 25.

26. 27.

Use a compass and a straightedge to bisect each line segment.

28. 29.

30. 31.

32. 33.

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34. 35.

Given the length of a line segment, calculate the measure of each new segment formed when the line is bisected.

36. ___

AB � 16 mm

37. ___

CD � 34 inches

38. ___

EF � 328 meters

39. ____

GH � 514 feet

40. __

IJ � 93 cm

41. ___

KL � 47 meters

42. __

IJ � 127 inches

43. ___

KL � 293 yards


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Use a compass and straightedge to draw the perpendicular bisectors of each side of a triangle. Extend the perpendicular bisectors to the point at which they intersect.

44. 45.

46. 47.

48. 49.

50. 51.

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Warehouse SpacePoints of Concurrency in Triangles

Vocabulary

Describe in your own words similarities and differences between each pair of terms.

1. incenter and orthocenter

2. altitude and median

3. centroid and circumcenter

4. concurrent lines and point of concurrency

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Problem Set Draw the incenter of each triangle.

5. 6.

7. 8.

9. 10.

11. 12.

Draw the circumcenter of each triangle.

13. 14.

15. 16.


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Name _____________________________________________ Date _________________________

17. 18.

19. 20.

Draw the centroid of each triangle.

21. 22.

23. 24.

25. 26.

27. 28.

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Draw the orthocenter of each triangle.

29. 30.

31. 32.

33. 34.

35. 36.


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Name _____________________________________________ Date _________________________

Answer the questions about points of concurrency. Draw an example to illustrate your answer.

37. For which type of triangle are the incenter, circumcenter, centroid, and orthocenter the

same point?

38. For which type of triangle are the orthocenter and circumcenter outside of the triangle?

39. For which type of triangle are the circumcenter and orthocenter on the triangle?

40. For which type of triangle are the incenter, circumcenter, centroid, and orthocenter all inside

the triangle?

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Documents

Skills Practice - Mater Academy Charter Middle / High School