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 $y = 10^\circ$.

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!