UK Secondary (7-11)
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Exterior angle sum and other calculations
Lesson

Exterior angle in a triangle

Lets have a look at the triangle image above. 

What relationships do we know exist?

We know that $y+z=180$y+z=180, because Adjacent angles on a straight line are supplementary (they add up to $180$180 degrees).

We also know that $w+x+y=180$w+x+y=180 because The angle sum of a triangle is $180$180 degrees.

Since both of these equations sum to $180$180, we know that they are both equal. This means that we have

$y+z=w+x+y$y+z=w+x+y.

Subtracting $y$y from both sides of the equation gives us the result

$z=w+x$z=w+x.

Have a look at where these angles are positioned on the triangle. 

This special relationship is used frequently in solving geometrical problems. We say formally that the: exterior angle of a triangle is equal to the sum of the two opposite interior angles. Can you see how this statement is a description of how the angles are positioned.

This applet can show you how this is the case.  Change the size of the triangle and then rotate the angles to fit in the external angle space. Watch this video if you would like to see this interactive in action -