The problem asks us to find the measure of angle B, $m\angle B$, in a cyclic quadrilateral ABCD. We are given that $m\angle B = (4x+9)^\circ$ and $m\angle D = (3x+3)^\circ$.

1. Problem Description

The problem asks us to find the measure of angle B, $m\angle B$, in a cyclic quadrilateral ABCD. We are given that $m\angle B = (4x+9)^\circ$ and $m\angle D = (3x+3)^\circ$.

2. Solution Steps

Since ABCD is a cyclic quadrilateral (all four vertices lie on a circle), the opposite angles are supplementary, which means their measures add up to $180^\circ$. Therefore,

$m\angle B + m\angle D = 180^\circ$

Substituting the given expressions for $m\angle B$ and $m\angle D$, we get

$(4x+9) + (3x+3) = 180$

Combining like terms, we have

$7x + 12 = 180$

Subtracting 12 from both sides, we get

$7x = 180 - 12$

$7x = 168$

Dividing both sides by 7, we find

$x = \frac{168}{7} = 24$

Now we can find $m\angle B$ by substituting $x=24$ into the expression for $m\angle B$:

$m\angle B = 4x + 9 = 4(24) + 9 = 96 + 9 = 105$

3. Final Answer

$m\angle B = 105^\circ$