Proof of ASP

Hi guys! Today I discovered my own proof for the Angle sum property of a triangle. (Or so I think)

The proof goes like this.

1) Take a triangle ABC. Draw an angle bisector AD to the side BC.

2) Let Angle BAD=Angle DAC=aa^{\circ} , Angle ABC=bb^{\circ} and Angle ACB = cc^{\circ}

3) Angle ADC= a+ba^ {\circ}+b^{\circ} [External angle=Sum of both interior angles]

4) Similarly, Angle ADB = a+ca^ {\circ}+c^{\circ}

5) 2a+b+c=1802a^{\circ}+b^{\circ}+c^{\circ}={180}^{\circ} [Angles on a straight line]

6) Angle A+Angle B + Angle C =180{180}^{\circ}

Thus proved.

Please leave your comments in the box below. Kindly tell me if this proof has already been derived by someone :)

Note by Mehul Arora
4 years ago

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I would argue that you can't actually "prove" this because the angle sum property is basically the same as the parallel postulate which is independent of the first four postulates. You can even have geometries where the angle sum property does not hold.

Mursalin Habib - 4 years ago

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So can we say, That this is valid for the "Commonly used" Geometry? :P

Mehul Arora - 4 years ago

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It is the characteristic axiom of Euclidean geometry. Other geometries that do not have this are called non-Euclidean geometries.

Mursalin Habib - 4 years ago

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Well, I had derived this as a solution to a textbook problem when I was of your age. By the way, good work.

Satyajit Mohanty - 4 years ago

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Thank you! :D

Mehul Arora - 4 years ago

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In step 3, isn't the fact that the external angle is the sum of the other two interior angles actually equivalent to the angle sum property for a triangle? You're saying 180ACD=a+b180^{\circ}-\angle ACD=a^{\circ}+b^{\circ} since they are the internal angles of a triangle, after all.

Can you alter your proof so that none of the steps assume that the sum of the internal angles of a triangle is 180180^{\circ}?

Maggie Miller - 4 years ago

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U used Exterior Angle Property which is proved by ASP. Hence this is wrong. :)

Rajdeep Dhingra - 4 years ago

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I assumed we know Exterior angle property :P

Mehul Arora - 4 years ago

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Assuming something in a proof is wrong.

Rajdeep Dhingra - 4 years ago

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@Rajdeep Dhingra I'm just joking :P

Mehul Arora - 4 years ago

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@Mehul Arora Hey come on Hangouts.

Rajdeep Dhingra - 4 years ago

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@Rajdeep Dhingra Okay :)

Mehul Arora - 4 years ago

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