Question

Based on the construction shown, what is the next step in the process of constructing parallel line AB? A straight line AB has a mid-point E such that CEB forms an acute angle α. Using the same compass width that created the first arc, repeat the arc centered at point C. Set the compass width to be where the arc intersects lines AB and CE. Using the same compass width that created the first arc, repeat the arc centered at point B. Set the compass width to the distance between point E and where the arc crosses line AB. a) Using the same compass width that created the first arc, repeat the arc centered at point C. b) Set the compass width to be where the arc intersects lines AB and CE. c) Using the same compass width that created the first arc, repeat the arc centered at point B. d) Set the compass width to the distance between point E and where the arc crosses line AB.

Answer 1

Set the compass width to the distance between point E and where the arc crosses line AB. Thus the correct option is D.

Step-by-step explanation:

This step involves setting the compass width to the distance between point E, the midpoint of line AB, and where the previously drawn arc intersects line AB. By adjusting the compass width to this measure, repeat the arc centered at point B. This action ensures the accurate replication of the arc's length and position from point B, contributing to the construction of the parallel line AB. This step maintains proportionality and alignment necessary for establishing parallel lines.

The process for constructing parallel lines using a compass and straightedge relies on maintaining congruence and proportionality between corresponding angles and segments. In this specific scenario, the initial steps established a midpoint E on line AB and created arcs from points C and B. To progress, aligning the compass width to the distance between E and the intersection of the previously drawn arc with line AB is crucial.

This measurement replicates the arc's length accurately when repeated from point B, ensuring the parallelism of line AB. Such precision in maintaining lengths and angles helps in constructing parallel lines geometrically. This method follows fundamental principles of geometry to ensure accuracy in line construction, leading to the establishment of the desired parallel line AB. Thus the correct option is D.

Answer 2

The next step in the process would be:

a) Using the same compass width that created the first arc, repeat the arc centered at point C.

To construct a line parallel to AB through point C using classical compass and straightedge construction, you would typically follow these steps:

1. Draw an arc across line AB and line segment EC using the same compass width. This step has already been completed as we can see from the construction shown.

2. Without changing the width of the compass, you then repeat this arc from point C to intersect the first arc. This will give you two points equidistant from AB, ensuring that the line drawn through these points will be parallel to AB.

Therefore, the next step in the process would be:

a) Using the same compass width that created the first arc, repeat the arc centered at point C.

This step will give you two new intersections: one on the line segment EC and one above it. A straight line through these two new points will be parallel to line AB.