The problem asks to find $m \angle ABD$ under the given conditions:

1. $\overline{BD}$ bisects $\angle CBE$,
2. $\overline{BC} \perp \overline{BA}$ (i.e., $\angle ABC = 90^\circ$),
3. $m \angle CBD = (3x + 25)^\circ$,
4. $m \angle DBE = (7x - 19)^\circ$.

Let's break this down step by step:

Step 1: Understanding the angle relationships

Since $\overline{BD}$ bisects $\angle CBE$, we know: $m \angle CBD = m \angle DBE$ This gives us the equation: $3x + 25 = 7x - 19$

Step 2: Solve for $x$

Solving the equation: $3x + 25 = 7x - 19$ Subtract $3x$ from both sides: $25 = 4x - 19$ Add 19 to both sides: $44 = 4x$ Divide by 4: $x = 11$

Step 3: Find the measures of $\angle CBD$ and $\angle DBE$

Substitute $x = 11$ into the expressions for $m \angle CBD$ and $m \angle DBE$: $m \angle CBD = 3(11) + 25 = 33 + 25 = 58^\circ$ $m \angle DBE = 7(11) - 19 = 77 - 19 = 58^\circ$ Thus, $m \angle CBE = m \angle CBD + m \angle DBE = 58^\circ + 58^\circ = 116^\circ$.

Step 4: Find $m \angle ABD$

Since $\overline{BC} \perp \overline{BA}$, we know that $\angle ABC = 90^\circ$. We are looking for $m \angle ABD$, and because: $m \angle ABC + m \angle ABD + m \angle DBC = 180^\circ$ Substitute the known values: $90^\circ + m \angle ABD + 58^\circ = 180^\circ$ Solve for $m \angle ABD$: $m \angle ABD = 180^\circ - 90^\circ - 58^\circ = 32^\circ$

Thus, $m \angle ABD = 32^\circ$.

Would you like more details or further clarification on any steps?

Here are five related questions:

1. How does the bisector of an angle affect its adjacent angles?
2. What is the significance of perpendicular lines in geometric angle calculations?
3. How do you solve linear equations involving angles in geometric figures?
4. What is the role of supplementary angles in solving for unknown angles in triangles?
5. How can angle bisectors be used to solve for unknown variables in geometry?

Tip: In geometry problems, clearly identifying known relationships like perpendicularity or bisectors can simplify the solution greatly.