To solve for $m \angle W$, we need to determine the measure of angle $W$ in the triangle $\triangle VWX$, given the following:

Known Information:

1. $\angle V$ (of $\triangle VWX$) = $72^\circ$.
2. In $\triangle BAC$, two angles are given: $\angle B = 31^\circ$ and $\angle A = 77^\circ$.
3. $m \angle W$ in $\triangle VWX$ needs to be calculated.

Steps:

1. Calculate the missing angle in $\triangle BAC$: The sum of the angles in a triangle is $180^\circ$. For $\triangle BAC$: $\angle C = 180^\circ - (\angle A + \angle B) = 180^\circ - (77^\circ + 31^\circ) = 72^\circ.$

2. Determine if $\triangle VWX$ is connected to $\triangle BAC$: Both triangles share the same angle $\angle W = \angle C = 72^\circ$.

Thus: $m \angle W = 72^\circ.$

Let me know if you'd like a detailed diagram or further clarifications. 😊

Follow-Up Questions:

1. What is the relationship between angles in two similar triangles?
2. How do you determine if two triangles are congruent or similar?
3. What is the sum of interior angles in any polygon?
4. How does the Angle Sum Property of triangles apply to quadrilaterals?
5. How can you use trigonometry to solve for angles in non-right triangles?

Tip: Always double-check angle relationships and triangle properties to ensure accuracy!