Name Class 3-2
Date Proving Lines Parallel
Connection: Constructing Parallel Lines
Essential question: How can you construct a line parallel to another line that passes
through a given point?
Video Tutor
Parallel lines lie in the same plane and do not
intersect. In the figure, line ℓ is parallel to line m
and you write ℓ ǁ m. The arrows on the lines also
indicate that the lines are parallel.
m
m
MCC9–12.G.CO.12
1
EXAMPLE
Constructing Parallel Lines
Construct a line parallel to line m that passes through point P. Work directly on the
figure below.
A
Choose points Q and R on line m.
P
P
Q
B
R
m
‹___›
Use a straightedge to draw PQ​
​    . 
m
C
Copy ∠PQR at point P, as shown. Label
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line ℓ. Line ℓ is the required line.
P
Q
P
R
m
Q
R
m
REFLECT
1a. Why does it make sense to copy ∠PQR to get a line parallel to line m?
1b. Is it possible to construct a line parallel to a given line m that passes through
a point P that is on line m? Why or why not?
Module 3
55
Lesson 2
1c. Write an if-then statement that justifies the method used in the construction of
parallel lines.
practice
Construct a line parallel to line m that passes through point P.
1.
2.
P
P
m
m
3.You can use what you know about constructing parallel lines to divide a given
line
___
segment into three equal parts. Follow the directions and work directly on AB​
​  
in the space below.
C
B
_​ __›
a. Use a straightedge to draw a ray AC​
​   as shown.
b. P
lace the point
of your compass on A and make an arc that
​___›
intersects AC​
​   . Without adjusting the
compass, place the point
​___›
on the intersection of the arc with AC​
​   and make another
_​ __arc.
›
Place the point on the intersection of the new arc with​_​AC​
__›   and
make a third arc. Label the points of intersection with AC​
​   
as shown.
A
B
X
Y
B
‹___›
___
Module 3
56
C
A
c. U
se
the straightedge to draw ​ZB​
    . Then construct lines parallel
___
___
to ZB​
​  that pass through X and Y. These lines divide ​AB​ into
three equal parts.
d. U
se a ruler to check that you have divided AB​
​  into three
equal parts.
Z
X
A
Y
Z
C
B
Lesson 2
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A
Name
Class
3-2
Date
Additional Practice
Construct a line parallel to line r that passes through point P.
1.
2.
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HJJG
3. Use the parallel line construction to divide CD into four congruent parts.
C
D
4. Construct a line parallel to AB through a point that is a distance XY
away from AB as measured along line m.
m
X
A
Module 3
Y
B
57
Lesson 2
Problem Solving
1. A cartographer is creating a map of the city center, as shown below.
Pine Street is parallel to Main Street and passes through the
intersection of Elm Street and Route 30. Construct a line to represent
Pine Street.
Route 30
Elm Street
Main Street
2. A graphic designer is drawing a logo, as shown below. In the logo,
AB = BC = DE = EF. Construct another line so that the logo is divided
into four rectangles that all have the same area.
D
B
E
C
F
Select the best answer.
4. When given a line m and a point P not on
m, what is the first step in constructing a
line parallel to m?
3. Which of the following facts do you use
when constructing parallel lines?
A A transversal that intersects two
parallel lines is parallel to both.
F Construct a circle with center P.
G Measure the distance between m
and P.
B Corresponding angles formed by two
parallel lines and a transversal are
congruent.
H Draw a line that intersects m and
contains P.
C Corresponding angles formed by two
parallel lines are supplementary.
J Draw a triangle with one side on m
and a vertex at P.
D All right angles are congruent.
Module 3
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Lesson 2
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A