Constructing and Bisecting a 105° Angle: A Comprehensive Guide

When dealing with angles in geometry, understanding how to construct and bisect specific angles is a fundamental skill. This guide will walk you through the process of constructing a 105° angle and bisecting it. We will cover the techniques using both traditional tools like compasses and rulers, and also introduce the use of dividers and protractors.

105° Angle Construction: A Step-by-Step Guide

To construct a 105° angle, you need to first understand how to construct 90° and 15° angles:

1. Constructing 90° Angle

To start with, the 90° angle can be easily constructed using a compass and a ruler.

Create a right angle initially by drawing a line (let's call it the x-axis) and then creating a perpendicular line (y-axis) at a point O, the origin. The angle formed between the x-axis and y-axis is a 90° angle. Using a compass, mark points C and D on the y-axis both above and below the x-axis at a distance r. This will ensure that CD 2r. Set the compass to measure a distance equal to 2r and center it at point O. Draw an arc in the bottom right quadrant. Draw a perpendicular line from D, intersecting the arc at point E. Connect O to E to form a 30° angle with the x-axis.

2. Constructing 30° Angle

To construct a 30° angle using a compass and straightedge:

Place the compass at O and draw an arc of radius r on the positive y-axis above the x-axis, marking point C, and another below the x-axis, marking point D. With the compass still set to the same radius r, place the compass at point C and draw an arc in the top right quadrant. Repeat the process from point D to draw another arc in the bottom right quadrant. Let the two new arcs intersect point E. This point will form a 30° angle with the x-axis. Draw a line from O to E, making the 30° angle.

3. Bisecting the 30° Angle

To bisect the 30° angle, follow these steps:

With the compass, draw arcs on the x-axis and OE, taking the same radius. Maintain the same radius and shift the compass pivot to the intersections of these arcs. Draw arcs that intersect at point F. Connect O to F to form a 15° angle with the x-axis.

4. Creating the 105° Angle

By combining the 90° and 15° angles, you can construct a 105° angle:

Since 105° 90° 15°, draw a 90° angle. Bisect the 30° angle to get a 15° angle. Add the 15° angle to the 90° angle to form a 105° angle.

Bisecting the 105° Angle

Bisecting the 105° angle follows the same process we used to bisect the 30° angle:

1. Bisecting with Compasses and Straightedge

Follow Steps 3 and 4 above to bisect the 30° and then combine to form the 105° angle, and then bisect it to form 52.5° angles.

2. Bisecting with Dividers, Protractor, and Ruler

If you prefer using dividers and a protractor:

Draw the line segment to be bisected, say 5 cm long. Set the dividers to a little past half the length, say 2.8 cm. From one end, draw two small arcs on each side of the line. From the other end, draw two more arcs crossing the first two. Connect the points of intersection with a straight line to bisect the line segment.

Additional Basic Tasks

Mastering these basic tasks will enhance your understanding of angle construction:

Constructing a right angle (90°). Constructing an equilateral triangle (60° angles at each vertex). Constructing a rhombus using compasses, where the diagonal is an angle bisector (or using a deltoid). Adding angles. Bisecting angles of 60° to get 15° and then combining them with 90° to form 105°.

Conclusion

Constructing and bisecting 105° angles is a crucial skill in geometry. By following the detailed steps outlined in this guide, you can ensure precision and accuracy in your constructions. Whether you prefer compasses, rulers, dividers, or protractors, the core principles remain the same. Master these techniques, and you'll be well on your way to becoming proficient in geometric constructions.