Question
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?
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?
Answers
Steve
PAD is not necessarily congruent to PBC.
They have two sides congruent, but that is not enough.
They have two sides congruent, but that is not enough.
Jonathan
PLEASE DO NOT POST PROBLEMS FROM THE PROMYS APPLICATION SET. YOU HAVE BEEN REPORTED.
Jeff
Wasn't it posted 11 months before your objection Jonathan??