Constructing Parallel Lines
Videos: Constructing Parallel Lines:

1. Begin with point P and line k. / 2. Draw an arbitrary line through point P, intersecting line k. Call the intersection point Q. Now the task is to construct an angle with vertex P, congruent to the angle of intersection. / 3. Center the compass at point Q and draw* an arc intersecting both lines. Without changing the radius of the compass, center it at point P and draw another arc. / 4. Set the compass radius to the distance between the two intersection points of the first arc. Now center the compass at the point where the second arc intersects line PQ. Mark the arc intersection point R.
5. Label the new line PR (or cursive l) and the two angles (example: 2 and 4). / / 6. Box a statement to show the two angles are congruent, corresponding 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 line k. / 2. Draw an transversal line through point P, intersecting line k. Label the intersection point Q. Now construct an angle with vertex P, congruent to one of the two exterior angles(1 or 2) at Q. / 3. Center the compass at point Q and draw* an arc intersecting both lines. Without changing the radius of the compass, center it at point P and draw another arc 180⁰ in the opposite direction (make sure they both pass thru the transversal). / 4. Span the compass radius to the distance between the two intersection points of the first arc. Now center the compass at the point where the second arc intersects transversal line PQ. Mark the arc intersection point R.
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P •
Q
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1 2 /
P •
Q
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1 2 / R
P •
Q
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1 2
5. Label the new line PR (or cursive l) and the two angles (example: 2 and 4). / R P
P •
Q
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1 2 / 6. Box a statement to show the two angles are congruent, Alternate Exterior Angles (AEA), so 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

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 k. (Make sure the point is farther away than is was in prior examples) / 2. Draw a transversal line through point P, intersecting line k. Label the intersection point Q. Now construct an angle with vertex P, congruent to one of the two interior angles (3 or 4) at Q. / 3. Center the compass at point Q and draw* an arc intersecting both lines. Without changing the radius of the compass, center it at point P and draw another arc 180⁰ in the opposite direction (make sure they both pass thru the transversal). / 4. Span the compass radius to the distance between the two intersection points of the first arc. Now center the compass at the point where the second arc intersects transversal line PQ. Mark the arc intersection point R.
P •
/ P •
3 4
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Q / P •
3 4
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Q / P •
3 4
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Q
5. Label the new line PR (or cursive l) and the two angles (example: 3 and 5). / P •
3 4
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Q / / 6. Box a statement to show the two angles are congruent, Alternate Interior Angles (AIA), so 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

Constructing a perpendicular to a line from a point OFF the line (“Dropping a Perpendicular”)

After doing this / Your work should look like this / After doing this / Your work should look like 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. /
1 / 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 /
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. / / 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) /
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. /

Constructing perpendicular from point ON a line (“Pulling” a perpendicular)

After doing this / Your work should look like 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 /
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

Advanced Construction (Both pairs of opposite sides are parallel using Alternate Interior Angles - AIA)

(1)Draw an original angle B / C
B 1
A
(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 / C

2

B 1 A
3
(4)Where the two lines from points C and A intersect is your final vertex. Label it “D” /
 ABCD
(5)Make a construction statement in a box showing that opposite sides are parallel.

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Do each Geometric Construction twice (using a compass and straight edge). Mark all new ANGLES, LINES, and congruent parts and box a statement describingthe 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.