We are given a quadrilateral $ABDC$ with angles $\angle B$ and $\angle C$ being right angles. We are also given that $m\angle BDC = (5x+39)^\circ$ and $m\angle A = (3x+13)^\circ$. We need to find the value of $x$ and the measure of angle $A$.

1. Problem Description

We are given a quadrilateral $ABDC$ with angles $\angle B$ and $\angle C$ being right angles. We are also given that $m\angle BDC = (5x+39)^\circ$ and $m\angle A = (3x+13)^\circ$. We need to find the value of $x$ and the measure of angle $A$.

2. Solution Steps

Since $ABDC$ is a quadrilateral, the sum of its interior angles is $360^\circ$. Therefore,

$$m\angle A + m\angle B + m\angle C + m\angle BDC = 360^\circ$$

We are given that $m\angle B = 90^\circ$ and $m\angle C = 90^\circ$. We also have $m\angle BDC = (5x+39)^\circ$ and $m\angle A = (3x+13)^\circ$. Substituting these values into the equation, we get:

$$(3x+13) + 90 + 90 + (5x+39) = 360$$

$$8x + 232 = 360$$

$$8x = 360 - 232$$

$$8x = 128$$

$$x = \frac{128}{8}$$

$$x = 16$$

Now, we can find $m\angle A$ by substituting $x=16$ into the expression for $m\angle A$:

$$m\angle A = (3x+13)^\circ$$

$$m\angle A = (3(16)+13)^\circ$$

$$m\angle A = (48+13)^\circ$$

$$m\angle A = 61^\circ$$

3. Final Answer

$x = 16$

$m\angle A = 61^\circ$