Proof of the Angle Sum Property

Why should the angles in any triangle sum to 180⁰? This question is answered by the proof of the angle sum property, which we are going to discuss now.

Consider an arbitrary triangle, \(\Delta {\rm{ABC}}\), as shown below:

We have to show that the sum of the angles x, y and z is 180⁰. We draw a line L through the vertex A, which is parallel to the side BC, as shown below:

Two additional angles are formed, which we have marked p and q. Now, we proceed to the proof.

Proof: Since AB is a transversal for the parallels L and BC, we have

p = y (alternate interior angles)

Similarly,  q = z. Now, p, x and q must sum to 180⁰ (why):

p + x + q = 180⁰

è y + x + z = 180⁰

Thus, the sum of the three angles x, y and z is 180⁰. And it should be obvious that this will hold true for any triangle, since the same proof is valid for any arbitrary triangle.