We are given a quadrilateral ABCD with the following angle measures: $\angle ABC = 14^{\circ}$, $\angle CDA = 31^{\circ}$, and $\angle DAB = 62^{\circ}$. We need to find the size of the reflex angle BCD.

1. Problem Description

We are given a quadrilateral ABCD with the following angle measures: $\angle ABC = 14^{\circ}$, $\angle CDA = 31^{\circ}$, and $\angle DAB = 62^{\circ}$. We need to find the size of the reflex angle BCD.

2. Solution Steps

First, we need to find the interior angle $\angle BCD$. We know that the sum of the interior angles of a quadrilateral is $360^{\circ}$. Therefore,

$\angle ABC + \angle BCD + \angle CDA + \angle DAB = 360^{\circ}$.

Substituting the given values, we have

$14^{\circ} + \angle BCD + 31^{\circ} + 62^{\circ} = 360^{\circ}$

$\angle BCD + 107^{\circ} = 360^{\circ}$

$\angle BCD = 360^{\circ} - 107^{\circ}$

$\angle BCD = 253^{\circ}$.

However, the angle we calculated is the reflex angle $\angle BCD$. To find the interior angle $\angle BCD$, we should calculate $360^{\circ}$ minus reflex angle.

Then we would have the following result:

$\angle BCD = 360^{\circ} - 253^{\circ}$

The problem is asking for the reflex angle BCD. Therefore,

The sum of angles in quadrilateral is

$14 + 31 + 62 + C = 360$, where C is the interior angle.

Then $107 + C = 360$.

$C = 360 - 107 = 253$.

Then the reflex angle is $360 - (360 - 107) = 253$.

3. Final Answer

253