The question provided describes a triangle $ABC$ with specific conditions for constructing it. Here’s a breakdown of the steps and how to solve it:

1. Given Conditions:

   • $m(\angle ABC) > 90^\circ$: This implies that triangle $ABC$ is an obtuse triangle.
   • $|AD| = |DC|$: Point $D$ is chosen on line segment $AC$ such that $AD = DC$, meaning $D$ is the midpoint of $AC$.
   • $|AB| = 10 \, \text{cm}$ and $|BC| = 6 \, \text{cm}$.
   • Line segment $BD \perp BC$, which implies that $BD$ is the height of the triangle from point $B$ perpendicular to $BC$.

2. Question:

   • The problem asks for the area of triangle $ABC$.

3. Solution:

   • Since $BD$ is the height from $B$ to the base $BC$, we can use the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
   • Here, $BC$ is the base with length $6$ cm, and $BD$ is the height.
   • Since the values for $AB$ and $BD$ are derived from an equilateral setup where triangle $ABD$ or $CBD$ would form an equal division, the area calculation results in a solution of 24 cm² based