According to the Journal of Irreproducible Results, any obtuse angle is a right angle!
x
A
P
B
C
D
Here is their argument. Given the obtuse angle x, we make a quadrilateral ABCD
with ∠DAB = x, and ∠ABC = 90◦
, and AD = BC. Say the perpendicular bisector
to DC meets the perpendicular bisector to AB at P. Then P A = P B and P C =
P D. So the triangles P AD and P BC have equal sides and are congruent. Thus
∠P AD = ∠P BC. But P AB is isosceles, hence ∠P AB = ∠P BA. Subtracting, gives
x = ∠P AD − ∠P AB = ∠P BC − ∠P BA = 90◦
. This is a preposterous conclusion –
just where is the mistake in the “proof” and why does the argument break down there?
The mistake in the "proof" lies in the assumption that the quadrilateral ABCD can be constructed as described.
The argument fails because it assumes that by constructing the quadrilateral ABCD with specific angle measures and side lengths, the perpendicular bisector of DC will intersect the perpendicular bisector of AB at point P. However, this assumption is false.
In reality, constructing the quadrilateral ABCD as described leads to a contradiction since the given conditions cannot hold simultaneously. Specifically, it is not possible for ∠DAB to equal x, ∠ABC to equal 90 degrees, and for AD to equal BC while still maintaining a valid quadrilateral shape.
Therefore, the argument breaks down due to the false assumption made during the construction of quadrilateral ABCD.