To solve the angles in this circle, let's start by analyzing the given information:

• $\angle DAO = 28^\circ$
• $\angle OCA = 24^\circ$
• $BD$ is a diameter, so $\angle BOD = 180^\circ$ (because the angle subtended by a diameter in a circle is always a right angle).

We need to find the following angles:

1. $\angle ADO$
2. $\angle ABC$
3. $\angle OCA$ (already given as $24^\circ$)
4. $\angle ACB$
5. $\angle OBC$
6. $\angle CAB$

Step 1: $\angle ADO$

• In triangle $\triangle ADO$, we know $\angle DAO = 28^\circ$ and $\angle AOD$ (because $AO$ is a radius, and thus $\angle AOD$ is 90° - $\angle ODA$ ).

We need to find $\angle ADO$. Since the angles in a triangle add up to $180^\circ$:

$\angle ADO + \angle DAO + \angle OAD = 180^\circ$

Given that $\angle OAD = 28^\circ$, we can find $\angle ADO$.

Step 2: $\angle ABC$

• $\angle ABC$ is subtended by the arc $AC$. Using the property that the angle subtended by an arc at the center is twice the angle subtended at the circumference, we can find $\angle ABC$.

Step 3: $\angle ACB$

• Similarly, $\angle ACB$ can be found using the properties of the angles in a circle, such as the fact that opposite angles of a cyclic quadrilateral sum to $180^\circ$.

Step 4: $\angle OBC$

• This angle can be found by considering triangle $OBC$ and using the known angles.

Step 5: $\angle CAB$

• $\angle CAB$ is the same as $\angle ABC$ since they subtend the same arc.

Let me calculate these angles for you.Based on the calculations:

• $\angle ADO = 90^\circ$
• $\angle ABC = 66^\circ$
• $\angle OCA = 24^\circ$ (as given)
• $\angle ACB = 24^\circ$
• $\angle OBC = 66^\circ$
• $\angle CAB = 66^\circ$

Would you like more details on any of these calculations, or do you have any other questions?

Related Questions:

1. What is the relationship between the angles subtended by the same arc at the center and the circumference of the circle?
2. How does the angle subtended by a diameter relate to the angles at the circumference?
3. Can you determine if a quadrilateral is cyclic if one angle is $90^\circ$?
4. How do you use the properties of cyclic quadrilaterals to determine unknown angles?
5. How does knowing that $BD$ is a diameter help in solving the problem?

Tip:

Remember that in a circle, the angle subtended by a chord at the center is twice the angle subtended at the circumference, which is a key property for solving such geometric problems.