Constructing a Parallel Line

Constructing a Parallel Line

How to construct a line that is parallel to the given line.

Goal

Constructing a Parallel Line: Goal

Goal:
Constructing the line
that is parallel to the given line
and that passes through the given point
which is not on the line.

How to Construct

Constructing a Parallel Line: Step 1

Draw a line that passes through
the given line and the given point.
(red line)

Constructing a Parallel Line: Step 2

Place the point of the compass
at the intersection of the lines.
(red point)

Draw an arc that passes through the lines.
(red arc)

Constructing a Parallel Line: Step 3

Without changing the radius of the compass,
place the point of the compass
at the given point.
(red point)

Draw an arc (red arc)
just like the previous arc.

Constructing a Parallel Line: Step 4

Place the point of the compass
at the intersection
of the blue line and the previous arc.
(red point)

The radius is the distance
from the red point
to the intersecting point
of the given line and the blue arc.
(red dashed arc)

Constructing a Parallel Line: Step 5

Without changing the radius of the compass,
place the point of the compass
at the intersection
of the blue line and the upper arc.
(red point)

Draw an arc
that passes through the upper arc.
(red arc)

Constructing a Parallel Line: Step 6

Draw a line that passes through
the given line and the intersection of the arcs.
(red line)

Then the red line is parallel to the given line.

Why the Constructed Line is Parallel to the Given Line?

The reason that the constructed line is parallel to the given line

What you did is constructing congruent angles.
(red angles)

So the red angles are corresponding angles
which are congruent.

So the constructed line
is parallel to the the given line.