Tell whether the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

We know that in a right angled triangle ,  the greatest side is
hypotenuse.

So if ABC  is a right angled triangle with a right angle
at B, then the greatest side is the hopetenuse AC .

Therefore AC^2 =
AB^2+BC^2.

Also from  trigonometry ,  for any angle
B,

AC^2  = AB^2+BC^2 - 2AB*BC
cosB....(1).

1 > CosB > 0  for B< 90
deg

CosB = 0 for B = 90 deg.

-1
< Cos B < 0 for 
B>0.

Therefore,

 AC^2  <
AB^2+BC^2 for  B < 90.

AC^2 = AB^2+BC^2  for B = 9
deg.

AC^2 > AB^2+BC^2 for B > 90
deg.

Therefore,  a triangle is acute , right or obtuse
according as the square on the greatest side is 
less than, or equal to, or greater than the sum of the
squares on the other two lesser
sides.