Calculate Angle BAD in an Obtuse Triangle with Perpendicular Height

To calculate $\angle \text{BAD}$, follow these steps:

• Step 1: Understand from the problem and diagram that $\angle ABC = 115^\circ$.
• Step 2: Recognize that $\angle ABD$ is supplementary to $\angle ABC$. Thus, $\angle ABD = 180^\circ - 115^\circ = 65^\circ$.
• Step 3: Use the right-triangle property in triangle $ADB$ (since $\angle ADB = 90^\circ$) to find $\angle BAD$.

Now, calculate $\angle BAD$ using the sum of angles in a triangle:

In triangle $ADB$, sum of angles is:

$\angle ADB + \angle BAD + \angle ABD = 180^\circ$

Substitute the known values:

$90^\circ + \angle BAD + 65^\circ = 180^\circ$

$\angle BAD = 180^\circ - 90^\circ - 65^\circ = 25^\circ$

Therefore, the size of angle $\angle \text{BAD}$ is $\boxed{25^\circ}$.