Question

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◦, and AD = BC. Say the perpendicular bisector to DC meets the perpendicular bisector to AB at P. Then PA = PB and PC = PD. So the triangles PAD and PBC 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?

Answer 1

the mistake is - the triangles PAD and PBC have equal sides and are congruent." The Δ PAD and ΔPAC have two equal sides but we know from geometry that congruent triangles require either equal side-side-side or angle-side-angle

Answer 2

Answer:

It is not about the side lengths.

Step-by-step explanation:

You are given at the start that AD = BC.

From drawing the bisectors, you can conclude that PA = PB and PC = PD.

This would mean that it follows the SSS rule for congruent triangles. And you can conclude that since ΔPAB is isosceles, then ∠PAB = ∠PBA and then conclude that x is 90°.

I have been given a diagram and can see that the angles do not match up but I have no clue how to figure out why