Download Construting parallel lines

Constructing Parallel Lines
Videos: Constructing Parallel Lines:
http://www.teachertube.com/video/constructing-parallel-lines-with-a-safety-com-228901
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
Constructing a perpendicular to a line from a point OFF the line (“Dropping a Perpendicular”)
After doing this
1
Your work should look like this
After doing this
Start with a line and
point R which is not
on that line.
2
Set the compasses'
width to a
approximately 50%
more than the
distance to the line.
The exact width
does not matter.
Place the compasses
on the given external
point R.
3
Draw an arc across
the line on each side
of R, making sure
not to adjust the
compasses' width in
between. Label
these points P and Q
Your work should look like this
4
At this point, you can
adjust the compasses'
width. Recommended:
leave it as is.
From each point P,Q,
draw an arc below the
line so that the arcs
cross.
5
Place a straightedge
between R and the
point where the arcs
intersect. Draw the
perpendicular line
from R to the line, or
beyond if you wish.
6
Done. This line is
perpendicular to the
first line and passes
through the point R.
It also bisects the
segment PQ (divides
it into two equal
parts)
Constructing perpendicular from point ON a line (“Pulling” a perpendicular)
After doing this
Start with a line and point K on that line.
1
Set the compasses' width to a medium setting. The
actual width does not matter.
2
Without changing the compasses' width, mark a
short arc on the line at each side of the point K,
forming the points P,Q. These two points are thus
the same distance from K.
3
Increase the compasses to almost double the width
(again the exact setting is not important).
4
From P, mark off a short arc above K
Your work should look like this
After doing this
5
Without changing the compasses' width repeat
from the point Q so that the the two arcs cross each
other, creating the point R
6
Using the straight edge, draw a line from K to
where the arcs cross.
7
Done. The line just drawn is a perpendicular to the
line at K
Your work should look like this
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
hw24
Do each Geometric Construction twice (using a compass and straight edge). Mark all new
ANGLES, LINES, and congruent parts and box a statement describing the construction.
Construct eight pairs of parallel line pairs, given a line and a point off the line.
Using Converse of CAP (twice: one acute /. one obtuse)
(1) Draw a line from the NW to SE and a point OFF the line above it to the NE
(2) Draw a transversal thru the line and the point approximately horizontal
(3) Pick one acute angle and copy it in the same direction onto the point off the line. (use the
transversal as your “image ray”)
Repeat, this time choosing the obtuse angle in step 3
Using Converse of AEAT (twice: one acute /. one obtuse)
(1) Draw a line from the NE to SW and a point OFF the line above it to the NW
(2) Draw a transversal thru the line and the point approximately horizontal
(3) Pick the EXTERNAL acute angle and copy it in the opposite direction onto the point off the
line. (use the transversal as your “image ray”)
Repeat, this time choosing the obtuse angle in step 3
Using Converse of AEAT (twice: one acute /. one obtuse)
(1) Draw a vertical line towards the left side of your paper and a point OFF the line far to the
right towards the SE
(2) Draw a transversal thru the line and the point going NW to SE
(3) Pick the INTERNAL acute angle and copy it in the opposite direction onto the point off the
line. (use the transversal as your “image ray”)
Repeat, this time choosing the obtuse angle in step 3
Using perpendicular transversals (twice: diagonal lines /. Horizontal and vertical lines)
(1) Draw a vertical line towards the left side of your paper and a point “P” OFF the line to the
right near its center.
(2) Drop a perpendicular from this point thru the line
(3) Pull another perpendicular from point P to the line you just drew in step 2
Repeat, this time starting with a diagonal line going NW to SE and the point OFF toward
the NE in step 1.