Is there a Pythagorean triple whose angles are 90, 45, and 45 degrees? Is there a Pythagorean triple (a.k.a. an integer triangle) whose angles are 90, 45 and 45 degrees? I am trying to connect LEGO roads at angles other than the standard 90 degrees.
 A: You cannot have an integer Pythagorean Triple whose angles are $45°, 45°$ and $90°$. 

Assume on the triangle we have sides $a$. Then by Pythagoras' Theorem,
$$a^2+a^2=2a^2=(a\sqrt{2})^2$$
This means the hypotenuse is no longer an integer length, because now it measures $a\sqrt2$. This means no such Pythagoren Triple exists.
A: No, since if the perpendicular sides are $a$ in length, the hypotenuse would be $a\sqrt2$. But $\sqrt2$ is irrational, so $a\sqrt2$ is not an integer.
A: In your context you might be interested in isosceles triangles that are almost right.  As others have said, a right isosceles triangle has sides that are $a,a,a\sqrt 2$ and as $\sqrt 2$ is not rational we cannot have an integer sided one.  However, if we find a rational number that is close to $\sqrt 2$ we can find isosceles triangles that are close to right.  We have $\sqrt 2 \approx 1.414213$, while $\frac 75 = 1.4$ is not so far away, so a $5,5,7$ triangle is close to right.  In fact the angle is $\arccos \left(\frac 1{50}\right)\approx 88.85^\circ$.  You might have enough give to tolerate that.  If not, given one triangle in the list is $a,a,b$ the next is $a+b,a+b,2a+b$, so the next is $12,12,17$, then $29,29,41$, and so on.  The get closer and closer to right as you progress.  If you are interested in where this comes from, you could look up Pell's equation.
