1. The problem involves finding the measure of angle $b$ in a triangle where the external angles at $a$ and $c$ are given as $65^\circ$ and $132^\circ$ respectively. 2. Recall that an external angle and its adjacent internal angle sum to $180^\circ$ because they form a linear pair. 3. Calculate the internal angle $a$: $$a = 180^\circ - 65^\circ = 115^\circ$$ 4. Calculate the internal angle $c$: $$c = 180^\circ - 132^\circ = 48^\circ$$ 5. Use the triangle angle sum property, which states that the sum of internal angles in a triangle is $180^\circ$: $$a + b + c = 180^\circ$$ 6. Substitute the known values: $$115^\circ + b + 48^\circ = 180^\circ$$ 7. Simplify and solve for $b$: $$b = 180^\circ - 115^\circ - 48^\circ = 17^\circ$$ 8. Therefore, the measure of angle $b$ is $17^\circ$.