We need to find the size of the reflex angle $BCD$ in the given figure. We have a triangle $ABD$ with angles $33^\circ$ at $D$, $51^\circ$ at $A$, and $25^\circ$ at $B$. We need to find the reflex angle at vertex $C$.

1. Problem Description

We need to find the size of the reflex angle $BCD$ in the given figure. We have a triangle $ABD$ with angles $33^\circ$ at $D$, $51^\circ$ at $A$, and $25^\circ$ at $B$. We need to find the reflex angle at vertex $C$.

2. Solution Steps

First, we calculate the angle $ADB$. The sum of the angles in the triangle $ABD$ is $180^\circ$.

$A + B + D = 180^\circ$

$51^\circ + 25^\circ + ADB = 180^\circ$

$76^\circ + ADB = 180^\circ$

$ADB = 180^\circ - 76^\circ = 104^\circ$

Then, we look for the angle $BCD$. We are given angle $BDC = 33^\circ$ and angle $DBC = 25^\circ$. We know that $BCD$ is exterior angle of triangle $BCD$, so

$BDA = BCD + DBC + BDC=58$

We are looking for angle $BCD$. Let's look at the diagram differently.

Let angle $BCA=x$ and angle $DCA = y$. The sum of the angles in triangle $ABC$ is 180, so

$x+51+25=180$, giving $x=180-76 = 104$.

The sum of the angles in triangle $ADC$ is 180, so

$y+51+33=180$, giving $y=180-84 = 96$.

Angle $BCD$ equals $x+y$. Therefore, the normal angle at $C = 104 + 96 = 200$. But the angle $200$ should be $360^\circ$ - x. However, instead we can write $BCD = 180^\circ - x$, where $x$ is the interior angle at vertex $C$.

The total angles around point $C$ is $360^\circ$. The normal angle at $C$ is $360$ - reflex angle at C.

To determine the angle $ACB$ and $ACD$, we can find the missing angles.

We know $\angle BAC = 51^\circ, \angle ABC = 25^\circ, \angle ADC = 33^\circ$.

$\angle ACB = 180 - 51 - 25 = 104^\circ$

$\angle ACD = 180 - 51 - 33 = 96^\circ$

The angle $BCD$ = $104+96 =200^\circ$.

$Reflex\,BCD = 360 - BCD$.

$Reflex\,BCD = 360 - (25 + 33 +51)=360-68=202$

$Reflex\,BCD = 360-200 = 160^\circ$

The angle at $C$ is $180 - 25 - 33 = 122$. So $Reflex=360-122=238$

Therefore, $\angle BCD$ is $25+33 = 58^\circ + X$ where X is the adjacent angle. The reflex angle should be $360^\circ - \angle BCD$ = $360 - 104 = 302$

Let's name the intersection $E$. In triangle $ABE$, $\angle AEB = 180 - (51 + 25) = 104$. Therefore $\angle DEC = 104$ as they are vertically opposite angles.

Then, in triangle $DEC$, the angle at C, $\angle DCE = 180 - (104 + 33) = 180 - 137 = 43$.

Therefore the reflex angle at C, is $360 - 43 = 317$.

3. Final Answer

317