Calculate Angle BAD: Quadrilateral with Algebraic Expressions x+3, 2x-2, 5x-2, and 2x+11

To solve this problem, we'll follow these steps:

• Step 1: Set up the equation for the sum of the angles of a quadrilateral.
• Step 2: Solve for the variable $x$.
• Step 3: Calculate angle $\angle BAD$ using the determined value of $x$.

Now, let's work through each step:

Step 1: We know that the sum of the interior angles in a quadrilateral is $360^\circ$. Therefore, we have:

$(x + 3) + (2x - 2) + (5x - 2) + (2x + 11) = 360$

Step 2: Simplify the equation:

$x + 3 + 2x - 2 + 5x - 2 + 2x + 11 = 360$

$10x + 10 = 360$

Solve for $x$ by subtracting 10 from both sides:

$10x = 350$

Divide both sides by 10:

$x = 35$

Step 3: Calculate $\angle BAD$ using $x = 35$:

$\angle BAD = x + 3 = 35 + 3 = 38^\circ$

Therefore, the solution to the problem is $\mathbf{38^\circ}$.