Question

a student is constructing a pair of parallel lines using a compass and ruler. here are his first steps:
step 1: the student draws his first line and labels it l. he marks a point a for the second line to pass through.
step 2: he uses a ruler to draw a line that passes through the point a and crosses the line l. he labels the point of intersection b.
step 3: he puts his compass point on b and makes an arc that crosses both of the lines. he labels the point where this arc intersects line l as c.
what should be his next step?
without changing the setting of the compass, put the compass point on a and draw an arc like the arc in step 3.
widen the compass to reach point a and draw another arc.
use his ruler to draw a straight line through point a that is parallel to line l.

Explanation:

Step1: Analyze the construction of parallel lines

When constructing parallel lines using a compass and ruler, after creating an arc from the intersection point \(B\) (Step 3), the next logical step is to replicate the arc - like construction from the other point \(A\) (the point through which the parallel line should pass) without changing the compass width (to maintain congruent angles). This is based on the principle of using corresponding angles (created by a transversal) to construct parallel lines.

Step2: Evaluate each option

• Option 1: Without changing the setting of the compass, put the compass point on \(A\) and draw an arc like the arc in Step 3. This is a valid step in the construction of parallel lines as it helps in creating congruent angles (by the transversal) which is a key part of the parallel - line construction.
• Option 2: Widen the compass to reach point \(A\) and draw another arc. Widening the compass disrupts the congruency of angles (which is required for parallel lines based on the corresponding - angles postulate).
• Option 3: Use his ruler to draw a straight line through point \(A\) that is parallel to line \(l\). Just using a ruler to draw a line through \(A\) without proper angle - congruency (using the compass for arc - based angle - copying) is not a valid geometric construction method for parallel lines.

Answer:

Without changing the setting of the compass, put the compass point on \(A\) and draw an arc like the arc in Step 3.