Let's break down the problem step by step using the given diagram:

• From the diagram, we have several angles labeled:
  • ∠B = 90° (right angle).
  • ∠C = 30°.
  • ∠D = 60°.
  • ∠A = 90° (right angle).

We need to calculate the unknown angles $a$, $x$, and $w$.

1. Finding angle $a$:

• In the triangle $ADE$, the sum of the interior angles is $180^\circ$.
• We already know two angles:
  • ∠D = 60°.
  • ∠A = 90°.

Using the angle sum property: $\angle a = 180^\circ - 90^\circ - 60^\circ = 30^\circ$

So, $a = 30^\circ$.

2. Finding angle $x$:

• In triangle $CDE$, the sum of the interior angles is also $180^\circ$.
• We know that:
  • ∠C = 30°.
  • ∠D = 60°.

Using the angle sum property: $\angle x = 180^\circ - 30^\circ - 60^\circ = 90^\circ$

So, $x = 90^\circ$.

3. Finding angle $w$:

• In triangle $CDE$, since we know two of the angles, we can directly calculate the third:
  • ∠C = 30°.
  • ∠E = 90° (since $x = 90^\circ$).

Using the angle sum property for triangle $CDE$: $\angle w = 180^\circ - 90^\circ - 30^\circ = 60^\circ$

So, $w = 60^\circ$.

Final Answers:

• $a = 30^\circ$
• $x = 90^\circ$
• $w = 60^\circ$

Let me know if you'd like more details or further clarifications.

Here are some related questions for you:

1. How do you apply the angle sum property in different types of triangles?
2. What is the significance of a 90° angle in geometry?
3. How do you use trigonometric ratios to find angles in triangles?
4. What are some real-life applications of calculating angles?
5. How do you determine unknown angles in a quadrilateral?

Tip: Always remember that the sum of interior angles of any triangle is 180°.