Find Angle BDC: Solving Quadrilateral with Expressions 3x-4, 2x+8, 6x+10, x-2

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

• Step 1: Set up the equation for the sum of the interior angles.
• Step 2: Solve for $x$.
• Step 3: Calculate $∠D$ using the value of $x$.

Now, let's work through each step:

Step 1: The angles of quadrilateral ABCD are given as follows:
$∠A = 2x + 8$, $∠B = 3x - 4$, $∠C = 6x + 10$, and $∠D = x - 2$.
According to the angle sum property of a quadrilateral, we have:

$(2x + 8) + (3x - 4) + (6x + 10) + (x - 2) = 360$

Step 2: Simplify and solve for $x$:

Combine like terms:
$2x + 3x + 6x + x + 8 - 4 + 10 - 2 = 360$

This simplifies to:
$12x + 12 = 360$

Subtract 12 from both sides:
$12x = 348$

Divide both sides by 12 to isolate $x$:
$x = 29$

Step 3: Substitute $x = 29$ into the expression for $∠D = x - 2$:
$∠D = 29 - 2 = 27$

Therefore, the measure of angle $∢\text{BDC}$ or $∠D$ is 27 degrees.