To solve for the measure of angle $\angle VWY$ in the given image, we need to analyze the information provided:

1. Identify the relevant angles:

   • Angle $\angle VWY$ is the angle formed between the rays $VW$ and $WY$.
   • There are also additional labeled angles in the image, which seem to provide complementary or supplementary information to find the required angle.

2. Use geometric properties:

   • The sum of the angles around a point is $360^\circ$.
   • The sum of the angles in a straight line is $180^\circ$.

Let me analyze the image and solve for $m \angle VWY$.Upon examining the image:

• It appears there are three vectors or rays: $\overrightarrow{VX}$, $\overrightarrow{VW}$, and $\overrightarrow{WY}$, emanating from point $V$.
• There is a right angle (marked as $90^\circ$) between $\overrightarrow{VX}$ and $\overrightarrow{VW}$.
• Another angle $\angle WVY$ is marked as $55^\circ$.

To find $m\angle VWY$, we need to consider the geometry:

1. Angles around a Point: The sum of angles around point $V$ on the same side is $180^\circ$.

$m\angle VWX + m\angle XWY + m\angle VWY = 180^\circ$

Given:

• $m\angle VWX = 90^\circ$ (right angle)
• $m\angle XWY = 55^\circ$

2. Calculate $m\angle VWY$:

$m\angle VWY = 180^\circ - (90^\circ + 55^\circ) = 180^\circ - 145^\circ = 35^\circ$

Conclusion:

The measure of $\angle VWY$ is $\boxed{35^\circ}$.

If you need more details or have further questions, please let me know!

Follow-Up Questions:

1. What is the sum of angles around a point, and how is it used in geometry?
2. How can we determine if angles are complementary or supplementary?
3. What other geometric properties can be used to solve for unknown angles?
4. How do we handle more complex geometric configurations involving multiple angles?
5. What is the significance of right angles in geometric proofs?

Tip:

Always check for right angles and straight lines in geometry problems; they often provide key clues for solving unknown angles.