Study Guide and Review
State whether each sentence is true or false. If false, replace the underlined word or number to make a
true sentence.
1. If
1
5, then lines p and q are skew lines.
SOLUTION: false; If
, then lines p and q are parallel lines.
2. Angles 4 and 6 are alternate interior angles.
SOLUTION: true
3. Angles 1 and 7 are alternate exterior angles.
SOLUTION: true
4. If lines p and q are parallel, then angles 3 and 6 are congruent.
SOLUTION: false; If lines p and q are parallel, then angles 3 and 6 are supplementary.
5. The distance from point X to line q is the length of the segment perpendicular to line q from X.
SOLUTION: true
6. Line t is called the transversal for lines p and q.
SOLUTION: true
7. If p
q, then
2 and
SOLUTION: False; If p q, then
8 are supplementary.
and
are congruent.
8. Angles 4 and 8 are corresponding angles.
SOLUTION: true
Classify the relationship between each pair of angles as alternate interior, alternate exterior,
corresponding, or consecutive interior angles.
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9. 1 and
5
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8. Angles 4 and 8 are corresponding angles.
SOLUTION: Study
Guide and Review
true
Classify the relationship between each pair of angles as alternate interior, alternate exterior,
corresponding, or consecutive interior angles.
9. 1 and
5
SOLUTION: and
lie on the same side of the transversal and on the same side of the other two lines. They are
corresponding angles.
10. 4 and
6
SOLUTION: and
are nonadjacent interior angles that lie on opposite sides of the transversal. They are alternate interior
angles.
11. 2 and
8
SOLUTION: and
are nonadjacent exterior angles that lie on opposite sides of the transversal. They are alternate exterior
angles.
12. 4 and
5
SOLUTION: and
are interior angles that lie on the same side of the transversal. They are consecutive interior angles.
13. BRIDGES The Roebling Suspension Bridge extends over the Ohio River connecting Cincinnati, Ohio, to Covington,
Kentucky. Describe the type of lines formed by the bridge and the river.
SOLUTION: The bridge and the river are represented by lines that do not intersect and are not coplanar. They are skew lines.
In the figure, m
used.
14. 1 = 123. Find the measure of each angle. Tell which postulate(s) or theorem(s) you
5
SOLUTION: In the figure, angles 1 and 5 are alternate exterior angles.
Alternate Exterior Angles Theorem:
If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.
.
By the Alternate Exterior Angles Theorem,
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15. 14
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13. BRIDGES The Roebling Suspension Bridge extends over the Ohio River connecting Cincinnati, Ohio, to Covington,
Kentucky. Describe the type of lines formed by the bridge and the river.
SOLUTION: Study
Guide and Review
The bridge and the river are represented by lines that do not intersect and are not coplanar. They are skew lines.
In the figure, m
used.
14. 1 = 123. Find the measure of each angle. Tell which postulate(s) or theorem(s) you
5
SOLUTION: In the figure, angles 1 and 5 are alternate exterior angles.
Alternate Exterior Angles Theorem:
If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.
.
By the Alternate Exterior Angles Theorem,
15. 14
SOLUTION: In the figure, angles 1 and 5 are alternate exterior angles, angles 5 and 13 are corresponding angles, and angles 13
and 14 form a linear pair.
.
By the Alternate Exterior Angles Theorem,
.
By the Corresponding Angles Postulate,
.
By the definition of a linear pair,
Substitute.
16. 16
SOLUTION: In the figure, angles 1 and 9 are corresponding angles and angles 9 and 16 form a linear pair.
By the Corresponding Angles Postulate, m∠9 = m∠1 = 123. Since angles 9 and 16 form a linear pair, they are supplementary.
17. 11
SOLUTION: In the figure, angles 1 and 5 are alternate exterior angles, and angles 5 and 11 are alternate interior angles.
By the Alternate Exterior Angles Theorem, m∠5 = m∠1 = 123.
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By the Alternate Interior Angles Theorem, m∠11 = m∠5 = 123 .
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Study Guide and Review
17. 11
SOLUTION: In the figure, angles 1 and 5 are alternate exterior angles, and angles 5 and 11 are alternate interior angles.
By the Alternate Exterior Angles Theorem, m∠5 = m∠1 = 123.
By the Alternate Interior Angles Theorem, m∠11 = m∠5 = 123 .
So, m∠5 = 123.
18. 4
SOLUTION: In the figure, angles 1 and 5 are alternate exterior angles and angles 4 and 5 form a linear pair.
Since alternate interior angles of parallel lines are congruent m∠1 = m∠5. Since m∠1 = 123, by substitution, m∠5 =
123.
Since angles 4 and 5 form a linear pair, they are supplementary.
Substitute.
19. 6
SOLUTION: In the figure, angles 1 and 3 are corresponding angles and angles 3 and 6 form a linear pair.
By the Corresponding Angles Postulate, m∠3 = m∠1 = 123.
Since a linear pair of angles is supplementary, m∠3 + m∠6 = 180.
Substitute.
20. MAPS The diagram shows the layout of Elm, Plum, and Oak streets. Find the value of x.
SOLUTION: By the Consecutive Interior Angles Theorem,
Solve for x.
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Determine whether
answer.
and .
Page 4
are parallel, perpendicular , or neither. Graph each line to verify your
Since a linear pair of angles is supplementary, m∠3 + m∠6 = 180.
Substitute.
Study Guide and Review
20. MAPS The diagram shows the layout of Elm, Plum, and Oak streets. Find the value of x.
SOLUTION: By the Consecutive Interior Angles Theorem,
Solve for x.
Determine whether
and answer.
21. A(5, 3), B(8, 0), X(–7, 2), Y(1, 10)
.
are parallel, perpendicular , or neither. Graph each line to verify your
SOLUTION: Substitute the coordinates of the points in slope formula to find the slopes of the lines.
The product of the slopes of the lines is –1. Therefore, the lines are perpendicular.
Graph the lines on a coordinate plane to verify the answer.
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22. A(–3, 9), B(0, 7), X(4, 13), Y(–5, 7)
SOLUTION: Page 5
By the Consecutive Interior Angles Theorem,
Solve for x.
Study Guide and Review
Determine whether
and answer.
21. A(5, 3), B(8, 0), X(–7, 2), Y(1, 10)
are parallel, perpendicular , or neither. Graph each line to verify your
SOLUTION: Substitute the coordinates of the points in slope formula to find the slopes of the lines.
The product of the slopes of the lines is –1. Therefore, the lines are perpendicular.
Graph the lines on a coordinate plane to verify the answer.
22. A(–3, 9), B(0, 7), X(4, 13), Y(–5, 7)
SOLUTION: Substitute the coordinates of the points in slope formula to find the slopes of the lines.
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Study Guide and Review
22. A(–3, 9), B(0, 7), X(4, 13), Y(–5, 7)
SOLUTION: Substitute the coordinates of the points in slope formula to find the slopes of the lines.
The two lines neither have equal slopes nor their product is –1. Therefore, the lines are neither parallel nor
perpendicular.
Graph the lines on a coordinate plane to verify the answer.
23. A(8, 1), B(–2, 7), X(–6, 2), Y(–1, –1)
SOLUTION: Substitute the coordinates of the points in slope formula to find the slopes of the lines.
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Study Guide and Review
23. A(8, 1), B(–2, 7), X(–6, 2), Y(–1, –1)
SOLUTION: Substitute the coordinates of the points in slope formula to find the slopes of the lines.
The two lines have equal slopes,
. Therefore, the lines are parallel.
Graph the lines on a coordinate plane to verify the answer.
Graph the line that satisfies each condition.
24. contains (–3, 4) and is parallel to
with A(2, 5) and B(9, 2)
SOLUTION: Find the slope of the line
The required line is parallel to
So, the slope of the required line is .
Start at (–3, 4). Move three units down and seven units right to reach the point (4, 1). Join the two points and extend.
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Study Guide and Review
Graph the line that satisfies each condition.
24. contains (–3, 4) and is parallel to
with A(2, 5) and B(9, 2)
SOLUTION: Find the slope of the line
The required line is parallel to
So, the slope of the required line is .
Start at (–3, 4). Move three units down and seven units right to reach the point (4, 1). Join the two points and extend.
25. contains (1, 3) and is perpendicular to
with P(4, –6) and Q(6, –1)
SOLUTION: Find the slope of the line
.
The required line is perpendicular to
. So, the slope of the required line is
Start at (1, 3). Move two units down and five units right to reach the point (6, 1). Join the two points and extend.
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26. AIRPLANES Two Oceanic Airlines planes are flying at the same altitude. Using satellite imagery, each plane’s
position can be mapped onto a coordinate plane. Flight 815 was mapped at (23, 17) and (5, 11) while Flight 44 was
Study Guide and Review
25. contains (1, 3) and is perpendicular to
with P(4, –6) and Q(6, –1)
SOLUTION: Find the slope of the line
.
The required line is perpendicular to
. So, the slope of the required line is
Start at (1, 3). Move two units down and five units right to reach the point (6, 1). Join the two points and extend.
26. AIRPLANES Two Oceanic Airlines planes are flying at the same altitude. Using satellite imagery, each plane’s
position can be mapped onto a coordinate plane. Flight 815 was mapped at (23, 17) and (5, 11) while Flight 44 was
mapped at (3, 15) and (9, 17). Determine whether their paths are parallel, perpendicular, or neither.
SOLUTION: Substitute the coordinates of the points in slope formula to find the slopes of the lines. Let
of the paths of flight 815 and flight 44 respectively.
and be the slopes The paths are parallel, since they have the same slope.
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Write an equation in point-slope form of the line having the given slope that contains the given point.
27. m = 2, (4, –9)
Study Guide and Review
26. AIRPLANES Two Oceanic Airlines planes are flying at the same altitude. Using satellite imagery, each plane’s
position can be mapped onto a coordinate plane. Flight 815 was mapped at (23, 17) and (5, 11) while Flight 44 was
mapped at (3, 15) and (9, 17). Determine whether their paths are parallel, perpendicular, or neither.
SOLUTION: Substitute the coordinates of the points in slope formula to find the slopes of the lines. Let
of the paths of flight 815 and flight 44 respectively.
and be the slopes The paths are parallel, since they have the same slope.
Write an equation in point-slope form of the line having the given slope that contains the given point.
27. m = 2, (4, –9)
SOLUTION: The point-slope form of a line is
where m is the slope and
is a point on the line.
Here, m = 2 and (x1, y 1) = (4, –9).
28. SOLUTION: The point-slope form of a line is y – y 1 = m(x – x1) where m is the slope and (x1, y 1) is a point on the line.
Here,
and (x1, y 1) = (8, –1).
So, the equation of the line is
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Write an equation in slope-intercept form of the line having the given slope and y-intercept.
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1
1
Study Guide and Review
28. SOLUTION: The point-slope form of a line is y – y 1 = m(x – x1) where m is the slope and (x1, y 1) is a point on the line.
Here,
and (x1, y 1) = (8, –1).
So, the equation of the line is
Write an equation in slope-intercept form of the line having the given slope and y-intercept.
29. m: 5, y-intercept: –3
SOLUTION: The slope-intercept form of a line of slope m and y-intercept b is given by y = mx + b.
Here,
and y-intercept = –3.
So, the equation of the line is
30. y-intercept: 4
SOLUTION: The slope-intercept form of a line of slope m and y-intercept b is given by y = mx + b.
Here,
and y-intercept = 4.
So, the equation of the line is
Write an equation in slope-intercept form for each line.
31. (–3, 12) and (15, 0)
SOLUTION: Use the slope formula to find the slope of the line.
Use the slope and one of the points to write the equation of the line in point-slope form.
The point-slope form of a line is
where m is the slope and
is a point on the line.
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Here,
,y ) =
and (x
1
1
So, the equation of the line is
(15, 0).
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The slope-intercept form of a line of slope m and y-intercept b is given by y = mx + b.
Here,
and y-intercept = 4.
Study
and Review
So,Guide
the equation
of the line is
Write an equation in slope-intercept form for each line.
31. (–3, 12) and (15, 0)
SOLUTION: Use the slope formula to find the slope of the line.
Use the slope and one of the points to write the equation of the line in point-slope form.
The point-slope form of a line is
where m is the slope and
is a point on the line.
Here,
and (x1, y 1) = (15, 0).
So, the equation of the line is
32. (–7, 2) and (5, 8)
SOLUTION: Use the slope formula to find the slope of the line.
Use the slope and one of the points to write the equation of the line in point-slope form.
The point-slope form of a line is
where m is the slope and
is a point on the line.
Here,
and (x1, y 1) = (5, 8).
33. WINDOW CLEANING Ace Window Cleaning Service charges $50 for the service call and $20 for each hour
Page 13
spent on the job. Write an equation in slope-intercept form that represents the total cost C in terms of the number
of
hours h.
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Study Guide and Review
33. WINDOW CLEANING Ace Window Cleaning Service charges $50 for the service call and $20 for each hour
spent on the job. Write an equation in slope-intercept form that represents the total cost C in terms of the number of
hours h.
SOLUTION: The slope-intercept form of a line with slope m and y-intercept b is given by y = mx + b.
Here,
and y-intercept = 50.
The total cost C in terms of the number of hours h is
.
Given the following information, determine which lines, if any, are parallel. State the postulate or theorem
that justifies your answer.
34. 7
10
SOLUTION: If angles 7 and 10 are congruent, than
; by the Converse of the Consecutive Interior Angles Theorem.
35. 2
10
SOLUTION: If angles 2 and 10 are congruent this does not imply any of the lines are parallel.
36. 1
3
SOLUTION: and
are corresponding angles of lines w and x.
Since
,
w || x by the Converse of Corresponding Angles Postulate.
37. 3
11
SOLUTION: and
are alternate exterior angles of lines v and z. Since
v || z by the Converse of Alternate Exterior Angles Theorem.
38. Find x so that
,
. Identify the postulate or theorem you used.
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37. 3
11
SOLUTION: and andare
alternate exterior angles of lines v and z. Since
Study Guide
Review
v || z by the Converse of Alternate Exterior Angles Theorem.
38. Find x so that
,
. Identify the postulate or theorem you used.
SOLUTION: Since
,
l || m by the Converse of Consecutive Interior Angles Theorem.
39. LANDSCAPING Find the measure needed for m ADC that will make
SOLUTION: By the Consecutive Interior Angles Theorem,
Substitute.
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if m BAD = 45.
.
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