Could someone me
in two or more complete sentences, explain the theorem used in solving for the range of possible lengths of the third side, ab of abc.

You can use the Pythagorean Theorem to find the length of the third side AB (Identified as "x" in the figure attached in the problem), which says that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs:

 a² = b²+c²

 As we can see the figure, the triangle does not have an angle of 90°, but it can be divided into two equal parts, leaving two triangles with a right angle. We already have the values of the hypotenuse and a leg in triangle "A" , so we can find the value of the other leg:

 b = √(a²-c²) b = √(10²-4²)
 b = 9.16

 With these values, we can find the hypotenuse in the triangle "B":
 x = √b²+c²
 x = √(9.16)²+(4)²
 x = 10