Question

Prove that in each triangle the total angles are 180 degrees

Answer 1

This is a proof that the angles in a triangle equal 180°:

The top line (that touches the top of the triangle) is
running parallel to the base of the triangle.

So:

angles A are the same angles B are the same

And you can easily see that A + C + B does a complete rotation from one side of the straight line to the other, or 180°

Answer 2

Let p be a line drown parallel to the side BC of ΔABC. 

Now, there are two parallel lines cut by a transversal. 

∡A = straight angle (180°) 

∡Y₂ + ∡Z + ∡X₂ = 180°

   l                  l                      ∡Y₁ ≡ ∡Y₂ (alternate interior angles)
   l                  l                      ∡X₁ ≡ ∡X₂ (alternate interior angles)      

∡Y₁ + ∡Z + ∡X₁ = 180°
                                                     

Note: See the attachment.