The problem asks to find the size of angle $a$ in the given quadrilateral. The other angles in the quadrilateral are $123^{\circ}$, $101^{\circ}$, and $85^{\circ}$.

1. Problem Description

The problem asks to find the size of angle $a$ in the given quadrilateral. The other angles in the quadrilateral are $123^{\circ}$, $101^{\circ}$, and $85^{\circ}$.

2. Solution Steps

The sum of the interior angles of a quadrilateral is $360^{\circ}$.

The formula for the sum of interior angles of a polygon with $n$ sides is:

$(n-2) \times 180^{\circ}$.

For a quadrilateral, $n=4$, so the sum of the interior angles is

$(4-2) \times 180^{\circ} = 2 \times 180^{\circ} = 360^{\circ}$.

Therefore, we have:

$123^{\circ} + 101^{\circ} + 85^{\circ} + a = 360^{\circ}$

$309^{\circ} + a = 360^{\circ}$

$a = 360^{\circ} - 309^{\circ}$

$a = 51^{\circ}$

3. Final Answer

The size of angle $a$ is $51^{\circ}$.