The image shows a geometric diagram with points $A$, $B$, $C$, and $D$, along with the question asking for the measure of $\angle ABD$. From the image:

• $\angle ABC = 90^\circ$ (since it's marked as a right angle).
• The angle between the line $BD$ and $BC$ is labeled $(4w)^\circ$.
• The angle between the line $AB$ and $BD$ is labeled $(7w - 9)^\circ$.

We can use the fact that the sum of the angles around point $B$ is $180^\circ$ (since they form a straight line).

Equation:

$\angle ABC + \angle CBD + \angle ABD = 180^\circ$ Substituting the known values: $90^\circ + 4w^\circ + (7w - 9)^\circ = 180^\circ$

Simplifying: $90 + 4w + 7w - 9 = 180$ $81 + 11w = 180$ $11w = 99$ $w = 9$

Now, substitute $w = 9$ back into the expression for $\angle ABD$: $\angle ABD = 7w - 9 = 7(9) - 9 = 63 - 9 = 54^\circ$

Thus, the measure of $\angle ABD$ is $54^\circ$.

Would you like more details on this process or have any questions? Here are some follow-up questions:

1. How do we determine the sum of angles around a point?
2. Why do angles on a straight line add up to $180^\circ$?
3. Can you explain how we solved for $w$ in the equation?
4. What does $\angle ABC = 90^\circ$ tell us about the triangle?
5. How would this change if the angles were in radians?

Tip: Always check if there is a right angle in a diagram, as it can simplify your calculations.