# Proof of the Angle Sum Property

Why should the angles in *any* triangle sum to 180^{0}? 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 180^{0}. 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 180^{0} (why):

p + x + q = 180^{0}

è y + x + z = 180^{0}

Thus, the sum of the three angles x, y and z is 180^{0}. And it should be obvious that this will hold true for *any* triangle, since the same proof is valid for any arbitrary triangle.