1. **State the problem:** We need to find the size of angle $a$ in a right triangle $PQR$ where \angle $P = 90^\circ$, \angle $R = 30^\circ$, and \angle $Q = 20^\circ$. Angle $a$ is an internal angle near point $P$ between sides $PR$ and $PQ$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles in any triangle is always $180^\circ$. 3. **Calculate the missing angle:** Since $P$ is a right angle, $\angle P = 90^\circ$. The other two angles given are $30^\circ$ at $R$ and $20^\circ$ at $Q$. However, these angles cannot all be interior angles of the same triangle because $90 + 30 + 20 = 140^\circ$, which is less than $180^\circ$. This suggests angle $a$ is not one of the main triangle angles but an internal angle formed at $P$ between $PR$ and $PQ$. 4. **Analyze angle $a$:** Since $P$ is $90^\circ$, and angle $a$ is inside the triangle near $P$, it must be complementary to the sum of the other angles at $P$ formed by the lines $PR$ and $PQ$. 5. **Use angle relationships:** The sum of angles around point $P$ on a straight line is $180^\circ$. Given the triangle and the angles, angle $a$ can be found by subtracting the known angles from $90^\circ$. 6. **Calculate angle $a$:** Since $\angle P = 90^\circ$, and the other angles at $P$ adjacent to $a$ are $20^\circ$ and $30^\circ$, angle $a = 90^\circ - 20^\circ = 70^\circ$. **Final answer:** $$a = 70^\circ$$