This is an extention of the pythagorean theorem.

lets take the two smaller side lengths, square them, and add them together. Then we will compare with the longest side squared to see if it fits the pythagorean theorem.

7^2 +10^2 =49+100=149

15^2=225

since 225 is not equal to 149 we have shown that this is not a right triangle. Well then what is it!?

if we look closely at the form a^2+b^2=c^2 we see that in our case the right side of the equation is greater than the left side. What this means is that the side that would have been the hypotenuse were this a right triangle is too long for it to be a right triangle. That means that the angle which would normally be a right angle must be greater than a right angle to create this side which is longer than what they hypotenuse would need to be. Therefore, it must be the case that this is an obtuse triangle.

Please note that if the part of the equation that is c^2 were less than the a^2+b^2 side then it would be acute.