Answer: C. The measure of angle H is $27$ ^{\circ}.

Explanation:

$1$ . Identify the properties of a right triangle:

   - A right triangle must have one interior angle that measures exactly $90$ ^{\circ}.

   - The sum of all interior angles in any triangle is always $180$ ^{\circ}.

$2$ . Analyze the given information:

   - We are given that $m\angle F = 63$ ^{\circ}.

   - For triangle FGH to be a right triangle, the sum of the other two angles, $m\angle G + m\angle H$ , must equal $180$^{\circ} - $63$^{\circ} = $117$^{\circ}. Additionally, one of the angles (F, G, or H) must be $90$^{\circ}.

$3$ . Evaluate the options:

   - A: The side length does not provide enough information about the angles to prove it is a right triangle.

   - B: The side length does not provide enough information about the angles to prove it is a right triangle.

   - C: If $m\angle H = 27$ ^{\circ}, we can find the measure of angle G.

     $m\angle G = 180$ ^{\circ} - $m\angle F$ - $m\angle H$

     $m\angle G = 180$ ^{\circ} - $63$^{\circ} - $27$^{\circ}

     $m\angle G = 180$ ^{\circ} - $90$^{\circ}

     $m\angle G = 90$ ^{\circ}

     Since angle G is $90$ ^{\circ}, the triangle is a right triangle. This information is sufficient.

   - D: The sum of the angles being $180$ ^{\circ} is true for ALL triangles, not just right triangles. It does not provide sufficient proof.

$4$ . Conclusion:

   - Knowing that the measure of angle H is $27$ ^{\circ} is sufficient to prove that triangle FGH is a right triangle. This matches option C.