We need to find the sizes of angles $a$ and $b$ in the given triangle. We are given the external angle of $a$ which is $278^{\circ}$, and another angle inside the triangle, which is $71^{\circ}$.

1. Problem Description

We need to find the sizes of angles $a$ and $b$ in the given triangle. We are given the external angle of $a$ which is $278^{\circ}$, and another angle inside the triangle, which is $71^{\circ}$.

2. Solution Steps

First, we need to find the value of angle $a$. We know that the sum of an interior angle and its corresponding exterior angle is $360^{\circ}$. So,

$a + 278^{\circ} = 360^{\circ}$

$a = 360^{\circ} - 278^{\circ}$

$a = 82^{\circ}$

Now that we have the value of $a$, we can find the value of angle $b$. We know that the sum of all interior angles in a triangle is $180^{\circ}$. Thus,

$a + b + 71^{\circ} = 180^{\circ}$

$82^{\circ} + b + 71^{\circ} = 180^{\circ}$

$153^{\circ} + b = 180^{\circ}$

$b = 180^{\circ} - 153^{\circ}$

$b = 27^{\circ}$

3. Final Answer

$a = 82^{\circ}$

$b = 27^{\circ}$