Question

Jack actually had two ways to trisect an angle. His second way is as follow the steps:

(1) Begin with an angle ∠ABC
. (2) Draw the line inside the angle that is parallel to BC and is one centimeter away from BC (the "centimeter" could be any unit of measurement - I’m using centimeter because it is easiest).
(3) Draw the line inside the angle that is parallel to AB and is two centimeters away from AB (the important point is that this distance is twice the other distance).
(4) The two drawn lines will intersect at a point, call it D.
(5) The angle ∠DBC is one third of ∠ABC. Of course, this is wrong. Do this construction on a 90◦ angle (on paper, or in Sketchpad), and then show that ∠DBC 6= 30◦. You can use trig if you want. And of course, measuring in Sketchpad is not a proof. Don’t measure the angle is Sketchpad, say "It’s not 30◦!", and expect credit for a proof.

Answer 1

Jack's second method of trisecting an angle by drawing two parallel lines inside the angle is incorrect.

Step-by-step explanation:

To trisect an angle, Jack's second method involves drawing two parallel lines inside the angle, with one at a distance of one centimeter from one side of the angle and the other at a distance of two centimeters from the other side. The point where the two parallel lines intersect is labeled as point D. The angle ∠DBC is claimed to be one-third of the angle ∠ABC. However, this construction is incorrect. We can demonstrate this by applying trigonometry to a 90◦ angle and showing that ∠DBC is not equal to 30◦.