According to the Journal of Irreproducible Results, any obtuse angle is a right angle! Here is
their argument.
Given the obtuse angle x, we make a quadrilateral ABCD with � DAB = x, and � ABC =
90◦, andAD = BC. Say the perpendicular bisector toDC meets the perpendicular bisector to
AB at P. ThenPA = PB andPC = PD. So the trianglesPADandPBC have equal sides
and are congruent. Thus � PAD = � PBC. But PAB is isosceles, hence � PAB = � PBA.
Subtracting, gives x = � PAD− � PAB = � PBC − � PBA = 90◦. This is a preposterous
conclusion – just where is the mistake in the “proof” and why does the argument break down
there?
It is advised to not cheat on the PROMYS application problems. If you are stuck on one then try another one.
The mistake in the "proof" lies in assuming that the quadrilateral ABCD can be constructed as described.
In the given argument, it is stated that AD = BC, which implies that the sides of the quadrilateral are equal in length. However, this assumption is incorrect. In a quadrilateral with an obtuse angle, the two sides adjacent to the obtuse angle will always be longer than the other two sides. Therefore, it is not possible to construct a quadrilateral ABCD with AD = BC and �DAB = x.
Because the initial assumption is false, the subsequent conclusions drawn in the argument are also incorrect. This means that the statement "x = 90◦" is not valid, and the argument breaks down at that point.
The mistake in the "proof" lies in assuming that the quadrilateral ABCD can be constructed as described. Let's break down the argument and identify the flaw:
The argument starts with the assumption of an obtuse angle x and constructs a quadrilateral ABCD with an angle of 90 degrees at vertex B and congruent sides AD and BC.
Next, it introduces point P, which is said to be the intersection of the perpendicular bisectors of DC and AB. It is claimed that PA = PB and PC = PD.
The argument then states that triangles PAD and PBC have equal sides and are congruent, leading to the conclusion that angle PAD equals angle PBC. It also claims that angle PAB equals angle PBA.
Finally, it subtracts angle PAB from angle PAD, as well as angle PBA from angle PBC, resulting in x = PBC - PBA = 90 degrees.
The flaw in this argument can be found in the assumption that a quadrilateral ABCD with the given properties can be constructed. In particular, it assumes that the perpendicular bisectors of DC and AB intersect at point P. However, this assumption is incorrect.
To see why, try drawing a diagram and constructing the quadrilateral as described. You will find that the perpendicular bisectors of DC and AB do not intersect, or they do intersect but not at point P. Therefore, the argument breaks down at this point because the subsequent calculations and conclusions rely on an incorrect assumption.