Problem 3 Classifying an Angle Pair STEM Architecture The photo below shows the Royal Ontario Museum in Toronto, Canada. Are angles 2 and4 alternate interior angles, same-side interior angles, corresponding angles, oralternate exterior angles? Angles 2 and 4 are alternate interior angles. TEKS Process Standard (1)(A) 1 2 3 4 PRACTICE and APPLICATION EXERCISES O N L I N E H O M E W O R K For additional support when completing your homework, go to PearsonTEXAS.com. Use the diagram to name each of the following. Assume that lines and planes that appear to be parallel are parallel. 1. a pair of parallel planes 2. all lines that are parallel to < AB > 3. all lines that are parallel to < DH > 4. two lines that are skew to < EJ > 5. all lines that are parallel to plane JFAE 6. a plane parallel to < LH > A E J C D H L F G B Scan page for a Virtual Nerd™ tutorial video. How do the positions of j2 and j4 compare? ∠2 and ∠4 are both interior angles and they lie on opposite sides of a line. 91 PearsonTEXAS.com
Transcript
Problem 3
Classifying an Angle Pair STEM
Architecture The photo below shows the Royal Ontario Museum in Toronto, Canada. Are angles 2 and 4 alternate interior angles, same-side interior angles, corresponding angles, or alternate exterior angles?
Angles 2 and 4 are alternate interior angles.
TEKS Process Standard (1)(A)
12
3
4
PRACTICE and APPLICATION EXERCISES
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Use the diagram to name each of the following. Assume that lines and planes that appear to be parallel are parallel.
1. a pair of parallel planes
2. all lines that are parallel to <AB
>
3. all lines that are parallel to <DH
>
4. two lines that are skew to <EJ>
5. all lines that are parallel to plane JFAE
6. a plane parallel to <LH
> A
E
J
CD
H
L
F G
B
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How do the positions of j2 and j4 compare?∠2 and ∠4 are both interior angles and they lie on opposite sides of a line.
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Identify all pairs of each type of angles in the diagram. Name the two lines and the transversal that form each pair.
7. corresponding angles
8. alternate interior angles
9. same-side interior angles
10. alternate exterior angles
Are the angles labeled in the same color alternate interior angles, same-side interior angles, corresponding angles, or alternate exterior angles?
11. 12. 13.
14. Apply Mathematics (1)(A) The photo shows an overhead view of airport runways. Are ∠1 and ∠2 alternate interior angles, same-side interior angles, corresponding angles, or alternate exterior angles?
15. Apply Mathematics (1)(A) You and a friend are driving go-karts on two different tracks. As you drive on a straight section heading east, your friend passes above you on a straight section heading south. Are these sections of the two tracks parallel, skew, or neither? Explain.
How many pairs of each type of angles do two lines and a transversal form?
26. a. Explain Mathematical Ideas (1)(G) Describe the three ways in which two lines may be related.
b. Give examples from the real world to illustrate each of the relationships you described in part (a).
d
e
8a
7
12 5
6
bc
43
54 1
36 2
3
45
6
12
34
5
6
1
2
1
2
A
E
J
I
C
D
H
F
GB
92 Lesson 3-1 Lines and Angles
TEXAS Test Practice
37. How many pairs of parallel planes does a cereal box have?
A. 2 B. 3 C. 4 D. 6
38. Construct MN congruent to XY .
X Y
27. Create Representations to Communicate Mathematical Ideas (1)(E) The letter Z illustrates alternate interior angles. Find at least two other letters that illustrate pairs of angles presented in this lesson. Draw the letters. Then mark and describe the angles.
28. Apply Mathematics (1)(A) A rectangular rug covers the floor in a living room. One of the walls in the same living room is painted blue. Are the rug and the blue wall parallel? Explain.
For Exercises 29–32, determine whether each statement is always, sometimes, or never true.
29. Two planes that do not intersect are parallel.
30. Two lines that lie in parallel planes are parallel.
31. Two lines in intersecting planes are skew.
32. A line and a plane that do not intersect are skew.
33. a. Connect Mathematical Ideas (1)(F) Suppose two parallel planes A and B are each intersected by a third plane C. Make a conjecture about the intersection of planes A and C and the intersection of planes B and C.
b. Find examples in your classroom to illustrate your conjecture in part (a).
Use the figure at the right for Exercises 34 and 35.
34. Do planes A and B have other lines in common that are parallel to
<CD
>? Explain.
35. Create Representations to Communicate Mathematical Ideas (1)(E) Are there planes that intersect planes A and B in lines parallel to
<CD
>? Draw a sketch to support your answer.
36. Create Representations to Communicate Mathematical Ideas (1)(E) A transversal r intersects lines / and m. If / and r form ∠1 and ∠2 and m and r form ∠3 and ∠4, sketch a diagram that meets the following conditions.
∠1 ≅ ∠2 ∠3 and ∠4 are supplementary.
∠3 is an interior angle. ∠2 and ∠4 lie on opposite sides of r.
∠4 is an exterior angle.
B
DC
A
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Which lines or segments are parallel? Justify your answer.
1. 2.
3. 4.
5. Justify Mathematical Arguments (1)(G) Complete the flow chart below.
Given: ∠1 and ∠3 are supplementary.
Prove: a } b
6. Apply Mathematics (1)(A) Two workers paint lines for angled parking spaces. One worker paints a line so that m∠1 = 65. The other worker paints a line so that m∠2 = 65. Are their lines parallel? Explain.
Analyze Mathematical Relationships (1)(F) Find the value of x for which O } m.
7. 8.
B
E G
C
S
P Q
T
M
C H
A45! 45!
R
J K
R S
L
Q T
M
a1 2
3 b
Def. of linear pair
∠1 and ∠2 aresupplementary.
a " b∠1 and ∠3 aresupplementary.
a. Supplements of thesame ∠ are ≅.
d.
c.
e.
b.
1
2 3
!
(3x ! 33)"
(2x # 26)"m
!
105!
m
(3x " 18)!
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104 Lesson 3-3 Proving Lines Parallel
Justify Mathematical Arguments (1)(G) Use the given information to determine which lines, if any, are parallel. Justify each conclusion with a theorem or postulate.
9. ∠2 is supplementary to ∠3.
10. ∠1 ≅ ∠3
11. ∠6 is supplementary to ∠7.
12. ∠9 ≅ ∠12
13. m∠7 = 65, m∠9 = 115
14. ∠2 ≅ ∠10
15. ∠1 ≅ ∠8
16. ∠8 ≅ ∠6
17. ∠11 ≅ ∠7
18. ∠5 ≅ ∠10
19. Write a paragraph proof.
Given: ∠2 is supplementary to ∠7.
Prove: / } m
20. Explain Mathematical Ideas (1)(G) If the rowing crew at the right strokes in unison, the oars sweep out angles of equal measure. Explain why the oars on each side of the shell stay parallel.
Analyze Mathematical Relationships (1)(F) Determine the value of x for which r } s. Then find mj1 and mj2.
1r
s 2
21. m∠1 = 80 - x, m∠2 = 90 - 2x
22. m∠1 = 60 - 2x, m∠2 = 70 - 4x
23. m∠1 = 40 - 4x, m∠2 = 50 - 8x
24. m∠1 = 20 - 8x, m∠2 = 30 - 16x
!
12
34
56
78
910
1112
m
a
b
Proof !
m32
67
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PRACTICE and APPLICATION EXERCISESON
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Find the slope of the line passing through the given points.
Use the given information to write an equation of each line.
10. slope 23, passes through (-2, -6) 11. slope -3, passes through (4, -1)
12. 13.
14. passes through (0, 5) and (5, 8) 15. passes through (6, 2) and (2, 4)
16. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Derive the slope formula. Include a graph as part of your answer.
For Exercises 17 and 18, write the equation of the horizontal and vertical lines though the given point.
17. (4, 7) 18. (0, -1)
19. Write equations for three lines that contain the point (5, 6).
Graph each pair of lines. Then find their point of intersection.
20. y = -4, x = 6 21. x = 0, y = 0 22. x = -1, y = 3
y
Ox
2
2
4!4 !2
(2, 2)
(!1, !4)O
y
64
4
!2!2!4
x
(!3, 4)
(3, !1)
y
Ox
2 64!2!4!6
2
!2
(!5, 3)
(3, 5)y
Ox
2!2!4
2
6(!2, 6)
(1, 3)
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23. What is the slope of the x-axis? Explain. Write an equation for the x-axis.
24. What is the slope of the y-axis? Explain. Write an equation for the y-axis.
25. Analyze Mathematical Relationships (1)(F) You want to construct a “funbox” at a local skate park. The skate park’s safety regulations allow for the ramp on the funbox to have a maximum slope of 4
11. If you use the funbox plan at the right, can you build the ramp to meet the safety regulations? Explain.
29. Apply Mathematics (1)(A) The equation P = - 133d + 1 represents the pressure P
in atmospheres a scuba diver feels d feet below the surface of the water.
a. What is the slope of the line? What does the slope represent in this situation?
b. What is the y-intercept (P-intercept)? What does it represent in this situation?
30. Analyze Mathematical Relationships (1)(F) The x-intercept of a line is 2, and the y-intercept is 4. Use this information to write an equation for the line.
31. Connect Mathematical Ideas (1)(F) The vertices of a triangle are A(0, 0), B(2, 5), and C(4, 0).
a. Write an equation for the line through A and B.
b. Write an equation for the line through B and C.
c. Compare the slopes and the y-intercepts of the two lines.
Do the three points lie on one line? Justify your answer.
35. AB has endpoints A(8, k) and B(7, -3). The slope of AB is 5. What is k?
A. 1 B. 2 C. 5 D. 8
36. Two angles of a triangle measure 68 and 54. What is the measure of the third angle?
F. 14 G. 58 H. 122 J. 180
37. One of the angles in a certain linear pair is acute. Your friend says the other angle must be obtuse. Is your friend’s conjecture reasonable? Explain.
128 Lesson 3-7 Equations of Lines in the Coordinate Plane
PRACTICE and APPLICATION EXERCISES
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For Exercises 1 and 2, are lines O1 and O2 parallel? Explain.
1. 2.
Write an equation of the line parallel to the given line that contains C.
3. C(0, 3); y = -2x + 1 4. C(6, 0); y = 13x
5. C(-2, 4); y = 12x + 2 6. C(6, -2); y = -3
2x + 6
For Exercises 7 and 8, are lines O1 and O2 perpendicular? Explain.
7. 8.
Write an equation of the line perpendicular to the given line that contains P.
9. P(6, 6); y = 23x 10. P(4, 0); y = 1
2x - 5 11. P(4, 4); y = -2x - 8
12. Apply Mathematics (1)(A) City planners want to construct a bike path perpendicular to Bruckner Boulevard at point P. An equation of the Bruckner Boulevard line is y = -3
4x. Find an equation of the line for the bike path.
Rewrite each equation in slope-intercept form, if necessary. Then determine whether the lines are parallel. Explain.
17. Analyze Mathematical Relationships (1)(F) Line /1 contains (-4, 1) and (2, 5), and line /2 contains (3, 0) and (-3, k). What value of k makes /1 and /2 parallel?
18. Connect Mathematical Ideas (1)(F) Write equations for two perpendicular lines that have the same y-intercept and do not pass through the origin.
19. Can the y-intercepts of two nonvertical parallel lines be the same? Explain.
y
Ox
2
1
4!6
(0, 3)
(4, 1)(!6, 1)
(0, !2)
!2!1
y
Ox
2
2
!1 4 6!6 !4
!1!2(!5, 1)
(1, 3)(4, 2)
(!4, !2)
y
Ox
2
5!2
!4
!4
(3, !3)
(!4, 4)
(1, !1)
(6, 3)
!1!2
y
Ox
4
4
!4
!4 (2, 0)
(!2, 4)
(!3, !2)
(3, 4)
!1!2
2 4P
Bruckner Blvd.
!2
3
!2!4
y
x
STEM
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132 Lesson 3-8 Slopes of Parallel and Perpendicular Lines
Use slopes to determine whether the opposite sides of quadrilateral ABCD are parallel.
24. Analyze Mathematical Relationships (1)(F) Are opposite sides of hexagon RSTUVW at the right parallel? Justify your answer.
25. Which line is perpendicular to 3y + 2x = 12?
A. 6x - 4y = 24 C. 2x + 3y = 6
B. y + 3x = -2 D. y = -2x + 6
Rewrite each equation in slope-intercept form, if necessary. Then determine whether the lines are perpendicular. Explain.
26. y = -x - 7 27. y = 3 28. 2x - 7y = -42
y - x = 20 x = -2 4y = -7x - 2
29. Apply Mathematics (1)(A) A community recently converted an old railroad corridor into a recreational trail. The graph at the right shows a map of the trail on a coordinate grid. The community plans to construct a path to connect the trail to a parking lot. The new path will be perpendicular to the recreational trail.
a. Write an equation of the line representing the new path.
b. What are the coordinates of the point at which the path will meet the recreational trail?
30. Justify Mathematical Arguments (1)(G) Is a triangle with vertices G(3, 2), H(8, 5), and K(0, 10) a right triangle? Justify your answer.
31. A triangle has vertices L(-5, 6), M(-2, -3), and N(4, 5). Write an equation for the line perpendicular to LM that contains point N.
O V U
R
TW
S
x
y
1 6
4
25
25
50
75
100
125
150
50 75 100 125 150O
x
y
O
Parking lot
Recr
eatio
nal T
rail
TEXAS Test Practice
32. △ABC is right with right angle C. The slope of AC is -2. What is the slope of BC?
33. What is the value of x in the diagram at the right?
34. The perimeter of a square is 20 ft. What is the area of the square in square feet?
