Step 1 :Given that line BH bisects angle EBC, we know that angle EBH equals angle HBC. Therefore, we can set up the equation \((6x+12)^{\circ} = (8x-10)^{\circ}\) and solve for x.
Step 2 :Solving the equation, we find that x equals 11.
Step 3 :Substituting x = 11 back into the expression for m angle EBH, we find that m angle EBH equals 78 degrees.
Step 4 :Therefore, the measure of angle EBH is \(\boxed{78}\) degrees.