1. **State the problem:** We are given a right triangle $\triangle ABC$ with a right angle at $C$ and an angle at $A$ measuring 70°. We need to find the measure of angle $B$.\n\n2. **Recall the triangle angle sum rule:** The sum of the interior angles in any triangle is always 180°. That is, \n$$m\angle A + m\angle B + m\angle C = 180^\circ.$$\n\n3. **Apply the known values:** Since $\angle C$ is a right angle, $m\angle C = 90^\circ$, and $m\angle A = 70^\circ$. Substitute these into the equation:\n$$70^\circ + m\angle B + 90^\circ = 180^\circ.$$\n\n4. **Simplify the equation:**\n$$70^\circ + 90^\circ + m\angle B = 180^\circ$$\n$$160^\circ + m\angle B = 180^\circ.$$\n\n5. **Solve for $m\angle B$:**\n$$m\angle B = 180^\circ - 160^\circ = 20^\circ.$$\n\n**Final answer:** \n$$m\angle B = 20^\circ.$$