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The problem shows a circle with the angles formed by chords and intersecting lines. Let's break this down to solve for \( x \) and \( y \):

### Step 1: Analyze the given angles.

We see:
- The angle marked \( 2x^\circ \) at \( T \),
- The angle marked \( 4y^\circ \) at \( U \),
- Two angles of \( 64^\circ \) at points \( S \) and \( V \).

### Step 2: Use the property of opposite angles in a cyclic quadrilateral.

A cyclic quadrilateral (one inscribed in a circle) has the property that opposite angles sum to \( 180^\circ \). Therefore, we can write the following equations:

\[
2x + 64^\circ = 180^\circ
\]
\[
4y + 64^\circ = 180^\circ
\]

### Step 3: Solve the equations.

#### For \( x \):
\[
2x + 64 = 180
\]
\[
2x = 180 - 64
\]
\[
2x = 116
\]
\[
x = \frac{116}{2} = 58^\circ
\]

#### For \( y \):
\[
4y + 64 = 180
\]
\[
4y = 180 - 64
\]
\[
4y = 116
\]
\[
y = \frac{116}{4} = 29^\circ
\]

### Final Answers:
\[
x = 58^\circ
\]
\[
y = 29^\circ
\]

Would you like further clarification on any step?

Here are 5 related questions to expand your understanding:
1. What is a cyclic quadrilateral, and why do its opposite angles sum to 180°?
2. How would the problem change if the angles given were external instead of internal?
3. Can we apply similar techniques to solve for missing angles in any polygon inscribed in a circle?
4. How would the solution differ if one of the angles given was obtuse?
5. What if instead of a circle, the shape were an ellipse—how would that affect angle calculations?

**Tip:** In geometry problems involving circles, always check for cyclic quadrilaterals, as they provide useful relationships between angles.

Find the values of x and y from the given angles inside a circle, with two angles marked as 64° and two unknown angles marked as 2x° and 4y°.

Learn how to solve for the unknown angles x and y in a cyclic quadrilateral where opposite angles sum to 180°. The problem features geometric principles like the angle sum property and is ideal for students in Grades 8-10.

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