We are given a triangle ABD with angle $D = 42^\circ$, angle $A = 51^\circ$, and angle $B = 15^\circ$. We need to find the size of the reflex angle BCD.

1. Problem Description

We are given a triangle ABD with angle $D = 42^\circ$, angle $A = 51^\circ$, and angle $B = 15^\circ$. We need to find the size of the reflex angle BCD.

2. Solution Steps

First, we need to find the angle ACB using the fact that angles in triangle ABC add to $180^\circ$.

$$\angle BAC = 51^\circ$$

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

Therefore,

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

Next, we need to find the angle CDB using the fact that angles in triangle ABD add to $180^\circ$.

$$\angle ADB = 42^\circ$$

$$\angle DAB = 51^\circ$$

Therefore,

$$\angle ABD = 180^\circ - (\angle ADB + \angle DAB) = 180^\circ - (42^\circ + 51^\circ) = 180^\circ - 93^\circ = 87^\circ$$

This value doesn't agree with $15^\circ$ given in the picture, because the picture shows that it should be a triangle ABC but not a triangle ABD. Therefore, there should be an angle ABC instead of angle ABD and we should use $15^\circ$ in the calculation of $\angle ACB$.

Now we need to find $\angle BCD$. We know that angles around a point add up to $360^\circ$. The reflex angle BCD is $360^\circ - \angle ACB$.

Therefore, the reflex angle BCD is $360^\circ - 114^\circ = 246^\circ$.

3. Final Answer

246 degrees