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=$a^{\circ}$ , Angle ABC=$b^{\circ}$ and Angle ACB = $c^{\circ}$

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

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

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

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

Thus proved.

Note by Mehul Arora
5 years, 5 months 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.

- 5 years, 5 months ago

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

- 5 years, 5 months ago

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

- 5 years, 5 months ago

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

- 5 years, 5 months ago

Thank you! :D

- 5 years, 5 months ago

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 $180^{\circ}-\angle ACD=a^{\circ}+b^{\circ}$ 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 $180^{\circ}$?

- 5 years, 5 months ago

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

- 5 years, 5 months ago

I assumed we know Exterior angle property :P

- 5 years, 5 months ago

Assuming something in a proof is wrong.

- 5 years, 5 months ago

I'm just joking :P

- 5 years, 5 months ago

Hey come on Hangouts.

- 5 years, 5 months ago

Okay :)

- 5 years, 5 months ago