https://file.aiquickdraw.com/mathbot/file/17295867978412yuu5nxu.jpg

Let's solve both problems shown in the image:

### Problem 1:
The first diagram involves a cyclic quadrilateral (inscribed in a circle) with the following angle measures:
- \( \angle E = 80^\circ \)
- \( \angle F = 70^\circ \)
- \( \angle G = 90^\circ \)
We need to find \( x \) and \( y \) (the angles \( \angle C \) and \( \angle D \)).

Since the quadrilateral is inscribed in a circle, we know that opposite angles of a cyclic quadrilateral are supplementary. That means:

1. \( \angle E + \angle G = 180^\circ \)
   \[
   80^\circ + 90^\circ = 170^\circ \quad \Rightarrow \quad y = 180^\circ - 170^\circ = 10^\circ
   \]

2. \( \angle F + \angle C = 180^\circ \)
   \[
   70^\circ + x = 180^\circ \quad \Rightarrow \quad x = 110^\circ
   \]

Thus, \( x = 110^\circ \) and \( y = 10^\circ \).

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### Problem 2:
The second diagram shows a right triangle inscribed in a circle, with:
- \( \angle a^\circ \)
- \( \angle b^\circ \)
- The third angle inside the triangle is \( 32^\circ \), and the right angle is at the center of the circle.

For a triangle inscribed in a circle with one angle subtended by a diameter, the angle opposite the diameter is \( 90^\circ \) (this is a property of the semicircle). Therefore:

1. Since the total sum of the angles in any triangle is \( 180^\circ \), we have:
   \[
   a + b + 32^\circ = 180^\circ
   \]
   
   Since \( a = 90^\circ \) (opposite the diameter):
   \[
   90^\circ + b + 32^\circ = 180^\circ \quad \Rightarrow \quad b = 58^\circ
   \]

Thus, \( a = 90^\circ \) and \( b = 58^\circ \).

---

Let me know if you'd like further clarifications or details! Here are 5 related questions:

1. How do the properties of cyclic quadrilaterals help solve angle problems in circles?
2. What are the properties of triangles inscribed in a circle?
3. Can you find the angles of any other polygons inscribed in a circle?
4. How would the solution change if the quadrilateral was not cyclic?
5. How do supplementary angles relate to inscribed shapes?

**Tip**: In any circle problem, remember that the sum of angles around a point or inside a polygon gives useful relations!

Find the values of x, y, a, and b from the two geometry problems with cyclic quadrilaterals and triangles inscribed in circles.

Learn how to find the angles in a cyclic quadrilateral and an inscribed triangle using circle theorems. This problem involves finding the values of x, y, a, and b, applying the inscribed angle theorem and the cyclic quadrilateral property.

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