We are given a quadrilateral with three known angles: $121^{\circ}$, $105^{\circ}$, and $82^{\circ}$. We need to find the measure of the fourth angle, which is labeled as $a$.

1. Problem Description

We are given a quadrilateral with three known angles: $121^{\circ}$, $105^{\circ}$, and $82^{\circ}$. We need to find the measure of the fourth angle, which is labeled as $a$.

2. Solution Steps

The sum of the interior angles of a quadrilateral is $360^{\circ}$. Therefore, we can write the equation:

$121^{\circ} + 105^{\circ} + 82^{\circ} + a = 360^{\circ}$

Combine the known angles:

$121 + 105 + 82 = 308$

So the equation becomes:

$308^{\circ} + a = 360^{\circ}$

To find $a$, subtract $308^{\circ}$ from both sides of the equation:

$a = 360^{\circ} - 308^{\circ}$

$a = 52^{\circ}$

3. Final Answer

The size of angle $a$ is $52^{\circ}$.