PARALLEL LINES

Two lines are said to be parallel if they are equidistant (equally distant) at all points.

Facts about parallel lines:

Two straight lines lying in the same plane either intersect or are parallel.

Through a point there can be only one parallel drawn to a given line.

If two lines are perpendicular to the third, and in the same plane, they are parallel.

BISECTING LINES

It is often necessary to find the midpoint of a line. This may be found by measuring, or by using dividers and finding it by trial and error. A much simpler way is by the use of a compass.

Example:

To bisect a line AB by using a compass:

Solution:

Step 1. Using A as a center and a radius more than 1/2 of AB, but less than AB, draw an arc.

Step 2. Using B as a center and the same radius as Step 1, draw an arc intersecting the arc drawn in Step 1. Mark intersecting points X and Y. Draw XY.

Conclusion:

AE = EB

NOTE: That E also represents the midpoint of XY and that XY is perpendicular to AB. XY is termed the perpendicular bisector of AB,

CONSTRUCTION OF PARALLEL LINES USING PERPENDICULARS

Example;

Construct parallel lines 2" apart.

Solution:

Step 1. Draw a base line and lay out two points A and B 2" apart.

Step 2. Construct perpendiculars AC and BD to AB at A and B.

Conclusion;

AC is parallel to BD.

Principle:

Perpendiculars to the same line are parallel.

NOTE: Horizontal parallel lines can be drawn by the same procedures.

DIVIDING LINES

Lines can be divided into equal parts by a number of methods. Four of these methods are (1) by using parallel lines, (2) by transferring angles, (3) by using equal segments on the side of an angle, and (4) by using a scale.

1. Using parallel lines

Example;

Divide AB into 5 equal parts.

Solution:

Step 1. Assume any angle ABD and draw BD.

Step 3. Assume a radius so that 5 times the radius will fall within the BD, Swing arcs using this radius on BD and AC.

Step 4. Connect B with the last arc swung from A and connect corresponding points.

Conclusion:

Lines drawn in Step 3 divide AB into 5 equal parts. 2. Transferring angles

Example:

Divide AB into 5 equal parts.

Solution:

Step 1. Draw a line AC at any assumed angle to AB.

Step 2. Step off with compass 5 equal parts on AC.

Step 3. At Q (the end of 5 parts), draw line BQ.

Conclusion:

Where sides of angles constructed in Step 4 meet AB, they will divide AB into equal parts.

3. Equal segments on the side of an angle

Example:

Divide AB into 5 equal parts.

Solution:

Step 1. At any assumed angle draw AC.

Step 2. Step off 5 equal parts on AC.

Step 3. At Q (the end of 5 parts), draw line BQ.

Step 4. Draw lines through P, 0, N, and M parallel to BQ.

Conclusion:

Where parallel lines intersect AB, AB will be divided into 5 equal parts.

Note the similarity of methods 3 and 2. 4. Use of a scale

Example:

Divide line AB, which is 29/16" long, into 6 equal parts.

Solution:

Step 1. At A draw a line perpendicular to AB.

Step 2. Place the scale at an angle so that the distance on the scale will divide easily into 6 parts. In the above, we have selected 3" which will divide into 6 equal parts of 1/2" each.

Step 3. Draw lines from 1/2"; each division is perpendicular to AB.

Conclusion:

The perpendiculars drawn will divide AB into 6 equal parts.

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