Often in a proof, it becomes helpful to modify the figure that you are given. It is never acceptable to change any of the original parts of the figure, but it is appropriate to draw in new lines that will help demonstrate something. Such lines are called auxiliary lines.

You can draw as many auxiliary lines into a figure as you want. The only thing you can't do is change or delete any of the existing parts of the figure. Some of the most common cases in which auxiliary lines are drawn are those in which you must attempt to show that two segments or angles are congruent. In these cases, it is often very helpful to draw in lines, or segments, that will create triangles. Then, by proving certain triangles congruent, you can show that their corresponding parts are also congruent, thus solving your problem and concluding your proof. The tricky part comes in deciding where to draw your lines so as to help in the simplest way, without making a mess of the given figure.

PARGRAPH Below is a sample proof in which an auxiliary line has been drawn. The auxiliary line is dotted to show which one it is.

Given: Triangle ABC

Prove: The angles of triangle ABC, angles, 1, 2, and 3, sum to 180 degrees.