https://file.aiquickdraw.com/mathbot/file/1726495615109czzoiuzo.jpg

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.

AB is the diameter with center O, and CD is parallel to BA. If ∠CAB = x = 80°, find: (i) ∠COB, (ii) ∠DAC, (iii) ∠DOC, (iv) ∠ADC.

Solve this circle geometry problem involving a diameter, parallel lines, and cyclic quadrilaterals. Given ∠CAB = 80°, find ∠COB, ∠DAC, ∠DOC, and ∠ADC using central and inscribed angle properties. Suitable for students in Grades 9-11.

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