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The problem involves a quadrilateral \( ABCD \) inscribed in a semicircle with \( BC \) as the diameter. You are asked to find the angle \( x \) and \( y \).

### Let's analyze the problem step by step:

1. **Given Information:**
   - Quadrilateral \( ABCD \) is inscribed in a semicircle.
   - \( BC \) is the diameter of the semicircle.
   - We need to find angles \( x \) and \( y \).

2. **Properties of the Inscribed Quadrilateral:**
   - Since \( ABCD \) is inscribed in a semicircle, \( \angle BAC = 90^\circ \) due to the fact that the angle subtended by a diameter in a semicircle is always a right angle.

3. **Identifying Angles \( x \) and \( y \):**
   - \( x = \angle ABC \)
   - \( y = \angle BCD \)

### Calculation of Angles:
   
#### 1. **Angle \( x \):**
   - Since \( \angle BAC = 90^\circ \) and considering the triangle \( ABC \), you can find \( x \) using the sum of angles in a triangle.
   \[
   \angle ABC + \angle BAC + \angle ACB = 180^\circ
   \]
   Since \( \angle BAC = 90^\circ \),
   \[
   x + \angle ACB = 90^\circ
   \]

#### 2. **Angle \( y \):**
   - In triangle \( BCD \), the sum of the angles will also be \( 180^\circ \):
   \[
   \angle BCD + \angle CBD + \angle DBC = 180^\circ
   \]
   Since \( \angle BCD = y \), we need more specific details or geometric relationships to solve for \( y \).

### Next Steps:
If you can confirm specific measurements or any additional relationships, we could calculate these angles more precisely.

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Do you want further details on this solution, or would you like to ask additional questions?

Here are some related questions to expand on this topic:

1. What is the relationship between the inscribed angle and the subtended arc in a semicircle?
2. How do the properties of cyclic quadrilaterals apply to this problem?
3. Can the Pythagorean Theorem help in calculating lengths in this problem?
4. What would change if the quadrilateral was not inscribed in a semicircle but in a full circle?
5. How can trigonometric identities assist in finding the angles \( x \) and \( y \)?

**Tip:** In any problem involving inscribed figures, always consider the properties of the angles subtended by the arcs and the relationships between the sides and angles in the figures.

ABCD is a quadrilateral inscribed in a semicircle with BC as its diameter. Find angles x and y.

Discover how to determine the angles x and y in a quadrilateral inscribed in a semicircle with BC as its diameter. This problem covers key geometric concepts, including the Inscribed Angle Theorem and Cyclic Quadrilateral Theorem, making it ideal for students in Grades 9-12.

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