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Thread: According to the Journal of Irreproducible Results, any obtuse angle is a right angl

  1. #1
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    According to the Journal of Irreproducible Results, any obtuse angle is a right angl

    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?
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    Re: According to the Journal of Irreproducible Results, any obtuse angle is a right a

    I suggest you draw it out. On my diagram \angle{ PAD} - \angle{PAB} \ne x
    Last edited by Archie; Mar 31st 2017 at 08:52 AM.
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    Re: According to the Journal of Irreproducible Results, any obtuse angle is a right a

    The error is here: "So the triangles PAD and PBC have equal sides and are congruent." The triangles PAD and PAC have two equal sides, but remember from geometry that congruent triangles require either equal side-side-side or angle-side-angle.
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    Re: According to the Journal of Irreproducible Results, any obtuse angle is a right a

    They have 3 equal sides.
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    Re: According to the Journal of Irreproducible Results, any obtuse angle is a right a

    Quote Originally Posted by Archie View Post
    They have 3 equal sides.
    Ah yes, that's right. OK, the problem then is with this statement: "Thus ∠PAD = ∠PBC" Angle PAD actually equals angle BPC, not PBC. So the sast substitution is incorrect.
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    Re: According to the Journal of Irreproducible Results, any obtuse angle is a right a

    Suffice to say that, having drawn an example out, I don't agree with that.
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    Re: According to the Journal of Irreproducible Results, any obtuse angle is a right a

    Having read this thread with great attention, I am ecstatic to learn that all obtuse angles are right angles.
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    Re: According to the Journal of Irreproducible Results, any obtuse angle is a right a

    Quote Originally Posted by JeffM View Post
    Having read this thread with great attention, I am ecstatic to learn that all obtuse angles are right angles.
    Perhaps he means the angle at the right of the quad
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    Re: According to the Journal of Irreproducible Results, any obtuse angle is a right a

    Thinking about this some more ...

    The error in the argument is in assuming that the line from P to D passes through line AB - that is, that PD crosses below A. But actually PD passes above A. This means that the angle PAD is not equal to x + PAB, but rather angle PAD = 360-(x+PAB). The rest of the "proof" then falls apart.
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