We are given a triangle ABD with angles $14^\circ$ at B, $62^\circ$ at A, and $31^\circ$ at D. We want to find the size of the reflex angle BCD.

1. Problem Description

We are given a triangle ABD with angles $14^\circ$ at B, $62^\circ$ at A, and $31^\circ$ at D. We want to find the size of the reflex angle BCD.

2. Solution Steps

First, we need to find the angle CBD. Since the sum of angles in triangle ABD is $180^\circ$, we have:

$14^\circ + 62^\circ + 31^\circ + \angle ADB = 180^\circ$

$\angle ADB = 180^\circ - (14^\circ + 62^\circ + 31^\circ)$

$\angle ADB = 180^\circ - 107^\circ = 73^\circ$

Therefore, $\angle ADB = 73^\circ$.

Next, we can find $\angle BCD$. The angles $\angle BCD$ and $\angle ADB$ must be supplementary, meaning they add to $180^\circ$. Thus,

$\angle BCD = 180^\circ - \angle ADB$

$\angle BCD = 180^\circ - 73^\circ = 107^\circ$

Finally, we need to find the reflex angle BCD. A reflex angle is an angle greater than $180^\circ$ but less than $360^\circ$. The reflex angle BCD is given by:

Reflex $\angle BCD = 360^\circ - \angle BCD$

Reflex $\angle BCD = 360^\circ - 107^\circ = 253^\circ$

3. Final Answer

The size of the reflex angle BCD is $253^\circ$.