Constructing Parallel Lines
Videos: Constructing Parallel Lines:
http://www1.teachertube.com/viewVideo.php?video_id=203474&title=Parallel_Lines_Construction
http://teachertube.com/viewVideo.php?video_id=120198&title=Constructing_Parallel_Lines
Given a line and a point, construct a line through the point, parallel to the given line using Corresponding Angles
(CA)
1. Begin with point P and
2. Draw an arbitrary line
3. Center the compass at
4. Set the compass radius to
line k.
through point P, intersecting
point Q and draw* an arc
the distance between the two
line k. Call the intersection
intersecting both lines.
intersection points of the first
point Q. Now the task is to
Without changing the radius
arc. Now center the compass
construct an angle with
of the compass, center it at
at the point where the second
vertex P, congruent to the
point P and draw another
arc intersects line PQ. Mark
angle of intersection.
arc.
the arc intersection point R.
5. Label the new line PR
6. Box a statement to show
(or cursive l) and the two
4
angles (example: 2 and 4).
l
the two angles are
congruent, corresponding
2
2  4, CA, l ǁ k
angles (CA), so line l is
parallel to line k.
*Note: these pair of angles can go in any of the four directions (in pairs) from the two vertices Q and P
Given a line and a point, construct a line through the point, parallel to the given line using Alternate Exterior Angles
1. Begin with point P and
2. Draw an transversal line
3. Center the compass at
4. Span the compass radius
line k.
through point P, intersecting
point Q and draw* an arc
to the distance between the
line k. Label the intersection
intersecting both lines.
two intersection points of the
point Q. Now construct an
Without changing the radius
first arc. Now center the
angle with vertex P,
of the compass, center it at
compass at the point where
congruent to one of the two
point P and draw another
the second arc intersects
exterior angles (1 or 2)
arc 180⁰ in the opposite
transversal line PQ. Mark the
at Q.
direction (make sure they
arc intersection point R.
both pass thru the
transversal).
R
P •
P •
Q
Q
•
1
5. Label the new line PR
k
2
R
P
(or cursive l) and the two
l
Q
•
1
2
Q
•
1
2
k
•
1
2
6. Box a statement to show
the two angles are
4
P •
angles (example: 2 and 4).
P •
congruent, Alternate
2  4, AEA, l ǁ k
Exterior Angles (AEA), so
k
line l is parallel to line k.
*Note: these pair of angles can be either the obtuse or acute angles (in pairs) from the two vertices Q and P
k
Given a line and a point, construct a line through the point, parallel to the given line using Alternate Interior Angles
1. Begin with point P and line
2. Draw a transversal
3. Center the compass at
4. Span the compass radius to
k. (Make sure the point is
line through point P,
point Q and draw* an arc
the distance between the two
farther away than is was in
intersecting line k.
intersecting both lines.
intersection points of the first
prior examples)
Label the intersection
Without changing the radius
arc. Now center the compass at
point Q. Now construct
of the compass, center it at
the point where the second arc
an angle with vertex P,
point P and draw another
intersects transversal line PQ.
congruent to one of the
arc 180⁰ in the opposite
Mark the arc intersection point
two interior angles (3
direction (make sure they
R.
or 4) at Q.
both pass thru the
transversal).
P •
P •
3
4
3
•
k
Q
P •
(or cursive l) and the two
3
•
Q
Q
3
k
4
•
Q
k
6. Box a statement to show the
l
5
P •
4
•
k
5. Label the new line PR
angles (example: 3 and 5).
P •
two angles are congruent,
Alternate Interior Angles (AIA),
4
3  5, AIA, l ǁ k
so line l is parallel to line k.
k
*Note: these pair of angles can be either the obtuse or acute angles (in pairs) from the two vertices Q and P
Sample homework assignements
Homework (A)
Construct eight parallel line pairs, given a line and a point off the line. Make
two each by constructing congruent angles for each pair of corresponding angles
(CA's) from each of the four directions from the vertex (upper right, lower
right, lower left, upper left). Make congruence statements for each. EC:
construct a set of three parallel lines using CA's.
Homework (B) See steps on prior pages
Construct four parallel line pairs, given a line and a point off the line. Make
four by constructing congruent angles for each pair of Alternate Exterior Angles
(AEA's) - doing both the obtuse and acute angles (two times each). Make
congruence statements for each construction.
Homework (B)
Construct eight parallel line pairs, given a line and a point off the line. Make
four by constructing congruent angles for each pair of Alternate Interior Angles
(AIA's) - doing both the obtuse and acute angles (two times each). Make two by
constructing supplementary angles for each pair of Same Side Interior Angles
(SSIA's). Make two more by constructing a perpendicular transversal thru the
point and then a new line perpendicular to the transversal at the point. Make
construction statements for each.
Advanced Construction (Both pairs of opposite sides are parallel using Alternate Interior Angles - AIA)
(1) Draw an original angle B
(2) Pick a random point along one side of angle B and label it
“C” and another random point A on the other side.
Note: You should choose the points farther away to avoid
crossing the measuring arcs.
(3) Copy the angle inside B (1) to the opposite direction to
point C (NE and SW along transversal BC) and to point A
(NE and SW along transversal BA)
(a) Make congruent measuring arcs in the same opposite
directions from each point
(b) Span the arc between the two sides of 1 and copy
this span onto the arc about point C and point A
(c) Extend a line from point C thru the intersection of the
measuring arc in (a) and the span in (b) and likewise
with the arcs about point A
(d) Label the new angle by points C and A as 2 and 3
(4) Where the two lines from points C and A intersect is your
final vertex. Label it “D”
(5) Make a construction statement in a box showing that
opposite sides are parallel.
C
B
1
A
C
2
B
D
1
A
3
1  2, AIA, AB ǁ CD
1  3, AIA, BC ǁ AD

ABCD