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 assuming that PAB is an isosceles triangle. Let's break down the argument step by step to understand why it is incorrect:

1. Given an obtuse angle x, a quadrilateral ABCD is constructed with ∠DAB = x and ∠ABC = 90◦, and AD = BC.

2. The perpendicular bisector of DC intersects the perpendicular bisector of AB at point P.

3. By construction, P A = P B and P C = P D.

4. It is incorrectly assumed that P AB is an isosceles triangle.

5. The conclusion is derived based on the assumption that P AB is isosceles: x = ∠P AD − ∠P AB = ∠P BC − ∠P BA = 90◦.

The argument breaks down because the assumption that P AB is isosceles is not justified. Without this assumption, we cannot conclude that x = 90◦. Therefore, the conclusion drawn from the argument is invalid.

In summary, the mistake in the "proof" lies in assuming that P AB is an isosceles triangle, which leads to the erroneous conclusion.

The mistake in the "proof" lies in assuming that the perpendicular bisector of AB and DC meet at point P. This assumption is incorrect, which leads to the erroneous conclusion that x is equal to 90 degrees.

To understand why the argument breaks down, let's analyze the situation step by step:

1. The argument starts with the assumption of an obtuse angle x.

2. Quadrilateral ABCD is constructed, where ∠DAB = x and ∠ABC = 90 degrees. It is also assumed that AD = BC.

3. The perpendicular bisector of DC and the perpendicular bisector of AB are introduced and assumed to intersect at point P.

4. It is claimed that PA = PB and PC = PD, which implies that triangles PAD and PBC are congruent because they have equal sides.

5. Consequently, it is inferred that ∠PAD = ∠PBC.

6. By considering the isosceles triangle PAB, it is claimed that ∠PAB = ∠PBA.

7. Subtraction of angles results in x = ∠PAD - ∠PAB = ∠PBC - ∠PBA = 90 degrees.

However, the mistake lies in assuming that the perpendicular bisectors of AB and DC intersect at point P. In reality, this assumption is not necessarily true for all obtuse angles x.

If we visualize the situation, we can see that for small obtuse angles, the perpendicular bisectors may intersect as described. However, as the angle x becomes larger, the perpendicular bisectors will not intersect at a single point. Therefore, the assumption made in the argument is invalid for larger obtuse angles.

In conclusion, the "proof" from the Journal of Irreproducible Results is flawed because it incorrectly assumes the intersection of perpendicular bisectors. The argument breaks down because the assumption is not valid for all obtuse angles, leading to the incorrect conclusion that any obtuse angle is a right angle.

It is an excellent exercise to follow instructions in order to draw a diagram.

I would let you do just that, and at any step the statement is incorrect, it is the required answer.
Hint: DC is not parallel to AB.

clarification:

Hint: DC is not parallel to AB unless ∠DAB is 90°.