ConstructionsModule 4 Lesson 1

Euclid created accurate geometric drawings using only a compass and a straightedge in ancient Greece. Architects still use these tools today to help create modern and amazing structures.

Constructing a line parallel to a given line through a point not on the line (congruent angle method)

Begin with a line and a point not on the line.

Use a straightedge to draw a line that intersects the original line and passes through the point.

Any angle formed by the two lines is fine.

Put the compass point on the intersection point of the two lines and draw an arc that passes completely through the angle formed.

With the same compass setting, move the compass point to the original point off the line and draw a similar arc that intersects the new line and is definitely below where the parallel line will be drawn.

Put the compass point on an intersection point of the first arc you drew and either line and open the compass to a width so that it passes through the other intersection point.

With the same compass setting, move the compass point to the intersection point of the second arc you drew and its line and draw an arc that intercepts the second arc you drew.

Align your straightedge with the original point not on the line and the last arc intersection point you drew and draw a line that passes through each.

The new line will be parallel to the original line.You will learn why these lines have to be parallel later in this module.

Here’s another method for constructing parallel lines given a line and a point not on the line. This is called the rhombus method.

Start with a line and a point not on the line.

Put the compass point on the point and open it wide enough to pass below the line.

Draw a long arc that passes through the line and past where the parallel line will be drawn. Keep this same compass setting.

With the same compass setting, place the compass point on the intersection of the arc and the line and draw an arc that intercepts the line.

Maintaining the same compass setting throughout, move the compass point to the intersection point of the line and the last arc your drew and draw an arc that intercepts the first arc you drew in this construction.

Position a straightedge to draw a line that intersects the original point and the intersection of the two arcs.

This new line must be parallel with the original line. You will learn why later in this course when studying parallelograms.

You created a special parallelogram called a rhombus in the process of this construction. You will explore rhombuses much more in Module 7.

Construct a line perpendicular to a given line that passes through a point not on the line. Begin with a line and a point not on the line.

Place the compass point on the point not on the line and open it wide enough to go beyond the original line. Then draw the two arcs that intercept the original line.

Place the compass point on one intersection point and open it past the midpoint of the two arcs and draw an arc below the original line.

Repeat the process with the compass point on the other intersection point. Make sure the arcs below the line intersect.

Use a straightedge to draw a line that intersects the original point and the intersection point of the final two arcs you drew.

You have now constructed a line perpendicular to a given line that passes through a point not on the line.

Next, we will construct a line perpendicular to a given line that passes through a point on the given line.

Start with a line and a point on the line.

Place the compass point on the given point on the line and open it to a reasonable width and draw two arcs that intercept the line on each side of the point.

Move the compass point to either intersection of an arc and the line and open it to a setting that is easily past the point.

Draw an arc either above or below the point. I am drawing above.

Without changing the compass setting, repeat the process with the compass point on the other intersection point.

Be sure this arc intercepts the arc you just drew.

Use a straightedge to draw a line that passes through the original point and the intersection of the final two arcs you drew.

This line must be perpendicular to the original line. They will meet to form a right angle.