We need to find the size of the reflex angle $BCD$ in the given triangle $ABD$. The angles at $A$, $B$, and $D$ are given as $51^{\circ}$, $15^{\circ}$, and $42^{\circ}$ respectively.

1. Problem Description

We need to find the size of the reflex angle $BCD$ in the given triangle $ABD$. The angles at $A$, $B$, and $D$ are given as $51^{\circ}$, $15^{\circ}$, and $42^{\circ}$ respectively.

2. Solution Steps

First, we need to find the angle $ABC$. The angles in a triangle add up to $180^{\circ}$.

So, in triangle $ABD$:

$\angle BAC + \angle ABD + \angle ADB = 180^{\circ}$

$51^{\circ} + 15^{\circ} + 42^{\circ} + \angle ACB + \angle BCD = 180^{\circ}$

However, we only know the external angles and that $A=51^{\circ}$, $B=15^{\circ}$, $D=42^{\circ}$. We want the reflex angle at C. We first need to find the size of $\angle ACB$.

The sum of the angles in triangle $ABD$ is $180^\circ$, so we have

$\angle BAC + \angle ABD + \angle ADB = 180^{\circ}$

$51^{\circ} + 15^{\circ} + 42^{\circ} = 108^{\circ}$

This is impossible because then the angles should be adding up to $180^{\circ}$. There must be another point in between $B$ and $D$ called $C$.

So, we are given $A = 51^{\circ}$, $B = 15^{\circ}$, and $D = 42^{\circ}$.

We want to find the reflex angle $BCD$. First we must find the angle $ACB$.

In $\triangle ABC$, $\angle BAC = 51^{\circ}$ and $\angle ABC = 15^{\circ}$.

Then $\angle ACB = 180^{\circ} - (51^{\circ} + 15^{\circ}) = 180^{\circ} - 66^{\circ} = 114^{\circ}$.

A reflex angle is an angle greater than $180^{\circ}$.

The reflex angle $BCD = 360^{\circ} - \angle ACB$, but we don't know what $\angle ACB$ is.

Let $\angle BCA$ be the interior angle at $C$. Then the reflex angle at $C$ is $360^{\circ} - \angle BCA$.

Since the sum of the angles in a triangle is $180^{\circ}$, the sum of the angles in triangle $ABD$ must be $180^{\circ}$. Therefore $\angle A + \angle B + \angle D = 51^{\circ} + 15^{\circ} + 42^{\circ} = 108^{\circ}$. $180^{\circ} - 108^{\circ} = 72^{\circ}$. Then we need to find $\angle BCD$ which is $360^{\circ} - 72^{\circ} = 288^{\circ}$.

Thus $\angle C = 180 - (51 + 15) = 180 - 66 = 114$.

Reflex angle BCD = $360^{\circ} - 114^{\circ} = 246^{\circ}$.

3. Final Answer

246