# Math Help - Geometry - Show that x+y+z=360

1. ## Geometry - Show that x+y+z=360

Ok. I've got a huge problem with this problem. Please use much text when you explain how to solve It so I understand.

2. Originally Posted by kevin3000
Ok. I've got a huge problem with this problem. Please use much text when you explain how to solve It so I understand.

$x = 180^0 - a$ etc.

and then use $a + b + c = 180^0$.

3. Using the exterior angle theorem $\left\{ \begin{gathered} x = b + c \hfill \\ y = a + b \hfill \\ z = a + c \hfill \\ \end{gathered} \right.$.

Note that $a+b+c=180$

4. Interestingly, the sum of the exterior angles of ANY polygon is also 360 degrees. Thinking informally, this is clear if you imagine rotating a pin aligned along one side so it aligns against the next side. To do that, you rotate it through the exterior angle. By the time you've done this for all exterior angles, you're back where you started and your pin has rotated through 360 degrees.

5. Just to throw my oar in, x and a, z and b, y and c each form a straight line so x+a= 180, z+ b= 180, and y+ c= 180. Adding those three equations, x+ a+ z+ b+ y+ c= 180+ 180+ 180= 540. but a+ b+ c= 180 because they are the interior angles in a triangle. 180+ x+ y+ z= 540 so x+ y+ z= 540-180= 360.