Proof of the Angle Sum Property

Proof of the Angle Sum Property

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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:

Arbitrary triangle

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:

Sum of angles of triangle

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.

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