One of the most beautiful aspects of geometry is how neatly everything fits together. Knowing how to bisect a line segment means we know how to make a angle. Knowing an equilateral triangle has three angles makes an easy job of constructing a angle with only a compass, a pencil, and a straightedge. Everything fits together. You need nearly the same tools Euclid used, thousands of years ago:
Connecting the arc intersection points with the straightedge will produce a neat, sure angle, with your transversal and the original line meeting at four right angles!
Something practically mystical surrounds . It forms interior angles of equilateral triangles. It is exactly 1/6th of a circle. Half of it plus itself forms a right angle. You could go on and on.
Making a angle begins by remembering that equilateral triangle.
The angle created from to to , by striking three congruent lengths, is . If you wanted, you could connect and and form the equilateral triangle. Hey, , you did it!
A angle is half of a angle. So, to draw a , construct a angle and then bisect it.
First, follow the steps above to construct your angle.
Bisect the angle with your drawing compass, like this:
Remember how we said everything in geometry fits together? What is the supplementary angle to an angle of ? In other words, what angle must we add to to get ?
Did you say ? Sure, a angle is the adjacent angle to any of the angles you already constructed!
To construct your angle, construct a angle and then extend one of its sides far past the vertex, like this:
[insert animation of 60° angle constructed, then run out the side and highlight the 120° angle adjoining it]
That angle beyond the angle is your angle.
None of your constructions required numbers or measurement. Euclid would be proud of you!
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