Analyze the image and extract the geometrical information. The image shows a circle with center O. Points A, B, C, and D are on the circle. Line segment AC is a diameter and passes through O. The angle ∠ABC is given as 130 degrees. There are also labels 'x' and 'y' indicating angles.

The image depicts a circle with center O and points A, B, C, and D on its circumference. AC is a diameter of the circle.

Given that ∠ABC = 130°.

In a circle, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. However, ∠ABC is an inscribed angle.

Consider the arc ADC. The angle subtended by this arc at the center is ∠AOC (which is 180° as AC is a diameter) and at the circumference is ∠ABC. This is incorrect as ∠ABC subtends arc ADC.

Let's re-examine the properties of inscribed angles and cyclic quadrilaterals.

ABCD is a cyclic quadrilateral. The sum of opposite angles in a cyclic quadrilateral is 180°.

Therefore, ∠ADC + ∠ABC = 180°.

This implies ∠ADC = 180° - 130° = 50°.

Also, ∠BAD + ∠BCD = 180°.

Since AC is a diameter, the angle subtended by the diameter at any point on the circumference is 90°. Thus, ∠ABC would be 90° if B was on the semicircle formed by AC. Since ∠ABC = 130°, B must be on the major arc formed by AC, which is not possible as AC is a diameter passing through O.

Let's assume the diagram is drawn such that ∠ABC is the reflex angle or there is a misunderstanding of the diagram. If we consider the angle subtended by arc ADC, it's ∠ABC = 130°. The angle subtended by arc ABC at the center is ∠AOC = 180°. The angle subtended by arc ADC at the circumference is ∠ABC = 130°.

There seems to be a contradiction in the provided information or diagram. Typically, an inscribed angle in a semicircle is 90°. If AC is a diameter, then ∠ABC should be less than or equal to 90° if B is on one side of the diameter. An angle of 130° for ∠ABC suggests it might be an angle measured in a different direction or that ABCD is not a simple convex cyclic quadrilateral as depicted.

Let's assume that 130° refers to the angle subtended by the major arc AC at point B. This would mean that the angle subtended by the minor arc AC at point B is 180° - 130° = 50°.

However, AC is a diameter, so it divides the circle into two semicircles. Any angle subtended by a diameter at the circumference is 90°.

Let's assume that the angle 130° is actually referring to an arc measure or a different angle. If we interpret ∠ABC as the angle subtended by arc ADC, then the measure of arc ADC is 2 * (180° - 130°) = 100° if it were a convex quadrilateral. This doesn't seem right.

Let's consider the possibility that the 130° is the measure of the arc ABC. Then the angle subtended at the center would be 130°. This is also unlikely given the diagram.

Let's assume the 130° is indeed ∠ABC as labeled.

Given that AC is a diameter, it implies that ∠ADC = 90° and ∠ABC = 90° if B and D were on the same semicircle. However, they are on opposite sides of AC.

If ABCD is a cyclic quadrilateral, then opposite angles sum to 180°.

So, ∠ADC + ∠ABC = 180°.

If ∠ABC = 130°, then ∠ADC = 180° - 130° = 50°.

Now consider triangle ADC. Since AC is a diameter, ∠ADC = 90°. This contradicts ∠ADC = 50°.

There is a fundamental inconsistency in the problem statement or the diagram. The given angle ∠ABC = 130° cannot exist if AC is a diameter and B is on the circle, because the angle subtended by a diameter at any point on the circumference is 90°.

Let's assume that 130° is actually the measure of the arc ADC.

If arc ADC = 130°, then the inscribed angle ∠ABC subtends this arc. However, inscribed angles are usually less than 180°.

Let's assume the 130° is the measure of the arc ABC. Then the inscribed angle ∠ADC subtends this arc. The measure of arc ABC = 130°. Then ∠ADC = 130°/2 = 65°.

If AC is a diameter, then arc ABC = 180° and arc ADC = 180°.

Let's consider the possibility that the 130° is an angle outside the triangle, for example, an angle related to a tangent, but there is no tangent.

Given the labels 'x' and 'y' for angles, let's assume that 'y' refers to ∠BAC and 'x' refers to ∠BCA.

In triangle ABC, if AC is a diameter, then ∠ABC should be 90°. Since it is given as 130°, there is an error in the problem statement or diagram.

Let's assume that the 130° is the angle subtended by the arc ADC at the center O, which is not possible since AC is a diameter and ∠AOC = 180°.

Assuming the diagram is correct and AC is a diameter:

1. ∠ABC = 90° (angle in a semicircle). This contradicts the given 130°.
2. If ABCD is a cyclic quadrilateral, ∠ADC + ∠ABC = 180°. If ∠ABC = 130°, then ∠ADC = 50°.
3. Since AC is a diameter, the angle subtended by AC at D is ∠ADC = 90°. This contradicts ∠ADC = 50°.

Due to the inconsistencies, a definitive solution cannot be provided based on the given information and standard Euclidean geometry principles.

However, if we are forced to interpret the diagram and markings as they are:

Let's assume that the 130° is not ∠ABC, but the reflex angle ∠ABC. This is unlikely.

Let's consider the arc subtended by ∠ABC. If ∠ABC = 130°, it subtends the major arc ADC. The measure of major arc ADC = 2 * (180° - angle at circumference). This is also confusing.

Let's assume that the diagram is misleading and AC is NOT a diameter, but just a chord. However, O is marked as the center and it lies on AC, so AC must be a diameter.

Let's ignore the