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?
Wasn't it posted 11 months before your objection Jonathan??
The mistake in the "proof" lies in assuming that the triangles PAD and PBC are congruent. The argument claims that since PA = PB and PC = PD, the triangles have equal sides, so they must be congruent. However, this is not a valid reasoning.
To understand why this assumption is incorrect, we need to analyze the properties of triangles. Two triangles are congruent if and only if their corresponding sides and angles are equal. In the given quadrilateral ABCD, we do not have any information about the angles or the lengths of the sides, except for the given angle measurements and the equality of AD and BC.
Since we cannot determine the lengths of the sides based solely on the given information, we cannot conclude that the triangles are congruent. As a result, the claim that ∠PAD = ∠PBC is not valid, and the argument breaks down there.
In geometry, it is essential to provide sufficient information to establish congruence or other relationships between geometric figures. In this case, without additional information, it is not possible to prove that an obtuse angle is a right angle solely based on the given quadrilateral ABCD.
PAD is not necessarily congruent to PBC.
They have two sides congruent, but that is not enough.