Constructing Parallel Lines

1
Constructing Parallel Lines

• Recall that parallel lines are two lines in the
  same plane that do not intersect.
• The Parallel Lines Conjecture states that if two
  lines cut by a transversal are parallel then the
  corresponding, alternate interior, or alternate
  exterior angles are congruent.
• The converse of the Parallel Lines Conjecture is
  also true
• If two lines are cut by a transversal and their
  corresponding angles, alternate interior angles,
  or alternate exterior angles are congruent, then
  the lines are parallel.
• This provides a way to construct parallel lines.

2
Constructing Parallel Lines

• Investigation 1
• Constructing Parallel Lines by Folding
• Step 1
• Draw a line and a point not on the line, as
  shown.
• Step 2
• Fold the paper to construct a perpendicular
  through the point. Draw a transversal along the
  crease.
• Step 3
• Through the point, make another perpendicular
  fold.
• Step 4
• Draw a second line through this crease.
• How can you confirm this line is parallel to the
  first line?

3
Constructing Parallel Lines

• Investigation 2
• Constructing Parallel Lines Alternate Method
• Step 1
• Construct a line on a piece of paper, and a point
  not on the line as before.
• Step 2
• Draw a transversal passing through the point and
  the first line you drew.
• Step 3
• How can you construct a parallel line using the
  angles made when the transversal crosses the
  first line?

4
Constructing Parallel Lines

• Parallel lines can be constructed using
• corresponding angles
• alternate interior angles
• alternate exterior angles
• perpendiculars to the same line
• These methods allow you to construct figures that
  have parallel sides, such as trapezoids,
  rhombuses, parallelograms, rectangles, and
  squares
• To construct parallel lines a specified distance
  apart, construct a perpendicular segment with the
  desired length from a point on the line.