Since the given is a right triangle, to solve for x, apply the Pythagorean formula,

`a^2+b^2=c^2`

where a and b are the legs of the triangle and c is the hypotenuse.

So, plug-in a=10, b=24 and c=x to the formula.

`10^2+24^2=x^2`

`100+576=x^2`

`676=x^2`

`+-sqrt576=sqrt(x^2)`

`+-26=x`

Since x represents the length of the hypotenuse, consider the positive value only.

Hence, the value of x is 26.

To solve for angle A, apply the tangent function.

`tan A = (opposite)/(adjacent)`

`tan A=24/10`

`tan A= 2.4`

`A=tan^(-1)2.4`

`A=67.38^o`

Thus, the measure of angle A is 67.38 degrees.

Since the sum of the three angles of the triangle is 180 degrees, to solve for B, add the three angles and set it equal to 180.

`A + B + 90^o = 180^o`

`67.3 8 ^ o +B+90^o=180^o`

`B+157.38^o=180^o`

`B=180^o-157.38^o`

`B=22.62^o`

Hence, the measure of angle B is 22.62 degrees.