1. **Stating the problem:** We have a right triangle ABC with a right angle at B and angle A measuring 55°. 2. **Understanding the angles:** In any triangle, the sum of the interior angles is 180°. 3. **Calculate angle C:** Since angle B is 90° and angle A is 55°, angle C is calculated as: $$\text{angle C} = 180^\circ - 90^\circ - 55^\circ = 35^\circ$$ 4. **Exterior angle at C:** The exterior angle at point C, formed by extending BC to D, is supplementary to angle C inside the triangle. This means: $$m = 180^\circ - \text{angle C} = 180^\circ - 35^\circ = 145^\circ$$ 5. **Final answer:** The measure of the exterior angle $m$ at point C is $145^\circ$.