Here is a famous false proof, apparently first
published by Lewis Carroll (who drew the diagram somewhat differently). I will
prove that a right angle is equal (congruent) to an arbitrary obtuse angle. Return to my Mathematics pages
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© Copyright 2001, Jim Loy
Here is a famous false proof, apparently first
published by Lewis Carroll (who drew the diagram somewhat differently). I will
prove that a right angle is equal (congruent) to an arbitrary obtuse angle.
Proof: We draw the quadrilateral ABCD in the diagram, so that angle A is the right angle, angle B is an arbitrary obtuse angle, and AC=BD. I will prove that angle A=angle B. We bisect segments AB and CD, and draw the perpendicular bisectors EG and FG (these perpendicular bisectors meet at some point G, as it is easy to show that they are not parallel). Draw the segments AG, BG, CG, and DG. By SAS (see Congruence Of Triangles, Part I), right triangle AGE is congruent to BGE, and CGF is congruent to DGF. So AG=BG, CG=DG, and angle GAE=angle GBE. So triangle AGC is congruent to triangle BGD, by SSS. So angle GAC=angle GBD. And in the quadrilateral, angle A=angle B (GAC-GAE=GBD-GBE). Which is what I intended to prove.
Of course, something is wrong with this proof, because a right angle is never equal to an obtuse angle. You might want to study the proof before reading further.
The flaw in the proof: On the right is a more accurate
diagram. In particular, notice that line GD does not ever intersect segment EB.
While we can still prove that the same long thin triangles are congruent, we
can never prove that angle A=angle B (in the quadrilateral).
So, the diagram was inaccurate. That is done all the time, and we seldom have to worry about it. We don't make all of our lines straight. And we may exaggerate some angles and segments. We normally do these things in order to make our diagrams clearer. And we don't want to make very accurate diagrams, and encourage students to measure the lines and think that this proves something. For example, we cannot prove the Pythagorean Theorem by measuring the lines with a ruler. And we don't draw a 3-4-5 right triangle and claim that this proves the theorem. In general, our proofs do not depend (much) on the accuracy of the drawing. But occasionally, an inaccurate diagram can lead us astray, as it does here.