Let's analyze the image and the given problem.

Problem Description:

• AB is the diameter of the circle, with O as the center.
• CD is parallel to BA (i.e., $CD \parallel BA$).
• $\angle CAB = x = 80^\circ$.

Required:

1. $\angle COB$
2. $\angle DAC$
3. $\angle DOC$
4. $\angle ADC$

Analyzing the Circle Geometry:

1. $\angle CAB = 80^\circ$: Since AB is the diameter, we can utilize the property of a cyclic quadrilateral and angles in the same segment.

2. Parallel lines $CD \parallel BA$: This provides information for using alternate angles and angle properties in the circle.

Now, let's solve each part step by step:

1. Finding $\angle COB$:

Since AB is the diameter, $\angle AOB$ (subtended by the diameter) is $180^\circ$. So, $\angle COB$ is the central angle subtended by arc $AB$.

$\angle COB = 2 \times \angle CAB = 2 \times 80^\circ = 160^\circ$

2. Finding $\angle DAC$:

Given that $CD \parallel BA$, $\angle DAC$ will be an alternate interior angle to $\angle CAB$.

$\angle DAC = \angle CAB = 80^\circ$

3. Finding $\angle DOC$:

Here, $\angle DOC$ is the central angle subtended by arc $DC$. We can use the fact that the central angle is twice the inscribed angle subtended by the same arc. Since $\angle DAC = 80^\circ$, we can compute:

$\angle DOC = 2 \times \angle DAC = 2 \times 80^\circ = 160^\circ$

4. Finding $\angle ADC$:

Since AB is the diameter, and D and C lie on the circle, we can use the property of angles subtended by the diameter. The inscribed angle subtended by the diameter is $90^\circ$. Hence, $\angle ADC$ is:

$\angle ADC = 90^\circ$

Final Answers:

1. $\angle COB = 160^\circ$
2. $\angle DAC = 80^\circ$
3. $\angle DOC = 160^\circ$
4. $\angle ADC = 90^\circ$

Would you like further clarifications or additional details?

Related Questions:

1. What is the sum of the angles in a cyclic quadrilateral?
2. How do parallel lines in a circle affect angles subtended by chords?
3. Can you explain why the angle subtended by a diameter is always $90^\circ$?
4. How would the problem change if $\angle CAB$ was given as $60^\circ$?
5. Can we derive the length of the arcs from these angle relationships?

Tip:

In circles, the angle subtended by the diameter at any point on the circumference is always a right angle (90°), which can simplify many problems involving cyclic quadrilaterals.