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Transcript
Constructing Parallelograms
There are five ways to construct a quadrilateral that is a parallelogram. The following are methods using a compass and straightedge to construct
a parallelogram using one of these five methods, based on having an original angle to start from.
1.
2.
3.
4.
5.
Corresponding Angles (CA’s)are congruent
Alternate Exterior Angles (AEA’s) are congruent
Alternate Interior Angles (AIA’s) are congruent
Two lines are perpendicular to the same line (perpendicular transversals)
Same Side Interior Angles (SSIA’s) are congruent (This construction in not shown, since it is drawn from CA’s)
Construction steps for #1 (Both pairs of opposite sides are parallel using Corresponding Angles - CA)
(1) Draw an original angle B
C
(2) Pick a random point along one side of angle B and label it
“C” and another random point A on the other side.
B
1
A
(3) Copy the angle inside B (1) to the same direction to
point C (corresponding angles along transversal BC) and
to point A (corresponding angles along transversal BA)
(a) Make congruent measuring arcs in the same direct
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
D
2
1
3
A
1  2, CA, AB ǁ CD
1  3, CA, BC ǁ AD

ABCD
Construction steps for #2 (Both pairs of opposite sides are parallel using Alternate Exterior Angles - AEA)
(1) Draw an original angle B
C
(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 must extend the lines past point B for each line
to copy the External Angles
(3) Copy the angle outside B (1) to the opposite direction to
point C (corresponding angles along transversal BC) and
to point A (corresponding angles along transversal BA)
(a) Make congruent measuring arcs in the same direct
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.
B
1
A
D
C
2
B
1
1  2, AEA, AB ǁ CD
1  3, AEA, BC ǁ AD
3
A

ABCD
Construction steps for #3 (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
Construction steps for #4 (Both pairs of opposite sides are parallel using Perpendicular Transversals)
(1) Draw an original angle B
(2) Pick a random point along one side of angle B and label it
“W”.
B
W
Note: You should choose the points farther away to avoid
crossing the semicircles
(3) Make a perpendicular transversal from point W
(a) Make semicircle about point W
(b) From each intersections for the semicircle to the line
WB make congruent arcs that intersect
(c) Extend a line from point W thru the intersection of the
arc in (b)
(4) Pick a random point on this transversal “X” and make
another perpendicular line from that point by repeating
steps (a) thru (c) above
(a) Make semicircle about point X
(b) From each intersections for the semicircle to the line
XW make congruent arcs that intersect
(c) Extend a line from point X thru the intersection of the
arc in (b)
Make
perpendicular
box
X
Make
perpendicular
box
B
W
(5) Extend the line from B to cross this parallel line (thru X),
Label this point C and Repeat steps (3) and (4) for this
line BC
(6) Pick a random point Y on BC Make a perpendicular
transversal from point Y
(a) Make semicircle about point Y
(b) From each intersections for the semicircle to the line
BC make congruent arcs that intersect
(c) Extend a line from point Y thru the intersection of the
arc in (b)
(7) Pick a random point on this transversal “Z” and make
another perpendicular line from that point by repeating
steps (a) thru (c) above
X
C
Y
B
Make
perpendicular
box
W
Z
Make
perpendicular
box
(8) Label the two points where this final line (perpendicular
to ZY, thru point Z) intersects the two parallel lines thru
points X and W as points A and D
X
C
D
Y
A
B
W
Z
(9) Make a construction statement in a box showing
that opposite sides are parallel
AB  WX, CD  WX so AB ǁ CD
BC  YZ, AD  YZ so BC ǁ AD

ABCD