We are given a triangle $ABD$ with an interior point $C$. We know the angles $\angle ADB = 42^\circ$, $\angle DAB = 51^\circ$, and $\angle DBC = 15^\circ$. We need to find the size of the reflex angle $BCD$.

1. Problem Description

We are given a triangle $ABD$ with an interior point $C$. We know the angles $\angle ADB = 42^\circ$, $\angle DAB = 51^\circ$, and $\angle DBC = 15^\circ$. We need to find the size of the reflex angle $BCD$.

2. Solution Steps

First, we find the angle $ABD$ in the triangle $ABD$. The sum of the angles in a triangle is $180^\circ$. So,

$\angle ABD = 180^\circ - \angle DAB - \angle ADB$

$\angle ABD = 180^\circ - 51^\circ - 42^\circ$

$\angle ABD = 180^\circ - 93^\circ$

$\angle ABD = 87^\circ$

Next, we find the angle $ABC$:

$\angle ABC = \angle ABD - \angle DBC$

$\angle ABC = 87^\circ - 15^\circ$

$\angle ABC = 72^\circ$

Now, we consider the triangle $ABC$. The sum of its angles is $180^\circ$. So

$\angle ACB = 180^\circ - \angle ABC - \angle BAC$

$\angle ACB = 180^\circ - 72^\circ - 51^\circ$

$\angle ACB = 180^\circ - 123^\circ$

$\angle ACB = 57^\circ$

The reflex angle $BCD$ is $360^\circ$ minus the angle $BCD$. We need to find the angle $BCD$. Since angles $ACB$ and $BCD$ form a straight line, the sum of those angles is $180^\circ$.

$\angle BCD = 180^\circ - \angle ACB$.

Substituting the value of $ACB$ we get

$\angle BCD = 180^\circ - 57^\circ$

$\angle BCD = 123^\circ$

Then we find the reflex angle $BCD$

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

Reflex $\angle BCD = 360^\circ - 123^\circ$

Reflex $\angle BCD = 237^\circ$

3. Final Answer

237