Question 945995
{{{x}}}= length of the third side.
To be a triangle, it must be
{{{10-5<x<10+5}}} ---> {{{5<x<15}}}
Within that range,
very short and very long third sides yield {{{red(obtuse)}}} triangles;
there are {{{x}}} values that yield right triangle, amd
in between those values we get acute triangles: 
{{{drawing(340,140,-6,11,-1,6,
green(arc(0,0,10,10,180,360)),
red(triangle(0,0,4.33,2.5,10,0)),
locate(2.17,1.25,red(5)),
triangle(2.89,4.08,0,0,10,0),
line(2.6,3.67,3.01,3.43),
line(3.01,3.43,3.3,3.84),
triangle(0,0,0,5,10,0),
rectangle(0,0,0.5,0.5),
red(triangle(0,0,-3.535,3.535,10,0)),
green(triangle(0,0,1.29,4.83,10,0)),
line(0,0,10,0),locate(5,0,10)
)}}}
{{{highlight(5<x<5sqrt(3)=about8.66)}}} or {{{highlight(5sqrt(5)<x<15)}}} --> obtuse triangle
{{{highlight(x=5sqrt(3)=about8.66)}}} or {{{highlight(x=5sqrt(5)=about11.18)}}} --> right triangle
{{{highlight(5sqrt(3)<x<5sqrt(5))}}} --> acute triangle
 
For {{{x=10}}} , we have an isosceles triangle,
with the vertex angle being the smallest angle, with measure {{{X}}} ,
because it is opposite the shortest side.
The other two angles are the base angles,
each measuring {{{(180^o-X)/2=90^o-X/2}}} ,
so they are acute too, and we have an acute triangle.
For The {{{x}}} values that yield right triangles, we use the Pythagorean theorem.
 
For {{{x>10}}}:
We would get a right triangle with {{{5}}} and {{{10}}} legs if and only if
{{{x^2=5^2+10^1}}})
{{{x=sqrt(5^2+10^2)=sqrt(25+100)=sqrt(125)}}}-->{{{highlight(x=5sqrt(5)=about11.18)}}}
If {{{x}}} is the longest side length, {{{x>10}}} ,
the largest angle would be opposite that longest side.
That angle would be greater than a right angle (an obtuse angle),
if and only if {{{x>5sqrt(5)=about11.18}}} .
So, we would get an obtuse triangle if and only if {{{highlight(5sqrt(5)<x<15)}}} .
Also with {{{x}}} being the longest side length, {{{x>10}}} ,
the largest angle, opposite that side, would be less than a right angle (acute) if and only if {{{10<x<5sqrt(5)=about11.18}}} 
So, we would get an acute triangle if {{{10<x<5sqrt(5)}}} ,
 
For {{{x<10}}}:
We would get a right triangle with {{{5}}} leg and {{{10}}} hypotenuse if and only if
{{{x^2+5^2=10^2}}}--->{{{x^2+25=100}}}
{{{x=sqrt(100-25)=sqrt(75)}}}--->{{{highlight(x=5sqrt(3)=about8.66)}}} .
When the longest side is the one with length 10,
the angle opposite that side is the greatest angle.
If {{{5<x<5sqrt(3)=about8.66)}}} ,
we would have {{{x^2+5^2<10^2}}} :
the greatest angle would be obtuse and the triangle would be obtuse.
If {{{5sqrt(3)<x<10)}}} ,
we would have {{{x^2+5^2>10^2}}} :
the greatest angle would be acute and the triangle would be acute.