To solve this problem, let’s analyze the geometry of the triangle:

1. The larger triangle is an isosceles triangle, as marked by the two equal sides.
2. The smaller right triangle splits the isosceles triangle into two parts.
3. We know:
   • The angle $w = 31^\circ$.
   • The right triangle implies that one of its angles is $90^\circ$.
   • $v, x, y$, and $w$ must satisfy the angle properties of both triangles.

Step-by-step solution:

1. Work with the right triangle:

• In a right triangle: $90^\circ + w + v = 180^\circ$ Substituting $w = 31^\circ$: $90^\circ + 31^\circ + v = 180^\circ \implies v = 59^\circ$

2. Angles of the isosceles triangle:

• For the larger triangle: The total interior angles must add up to $180^\circ$, and the two base angles (here $x$ and $y$) are equal since it’s isosceles. $w + x + y = 180^\circ$ Substituting $w = 31^\circ$: $31^\circ + x + x = 180^\circ \implies 2x = 149^\circ \implies x = 74.5^\circ$ Since $x = y$: $y = 74.5^\circ$

Final Answer:

• $w = 31^\circ$
• $v = 59^\circ$
• $x = 74.5^\circ$
• $y = 74.5^\circ$

Let me know if you'd like further clarification or details! Here are some questions to deepen your understanding:

5 Related Questions:

1. How do you calculate angles in an isosceles triangle given one base angle?
2. What are the properties of right triangles used in this solution?
3. How would the solution change if the angle $w$ was different?
4. Can you derive the formula for the vertex angle in an isosceles triangle?
5. How do the interior angles of triangles relate to the classification of triangles?

Tip:

In an isosceles triangle, the two base angles are always equal, and the vertex angle can be calculated as $180^\circ - 2 \times \text{base angle}$.