To solve for the angles $z$, $x$, and $y$ in the given diagram, we can use several geometric properties of circles, specifically related to angles subtended by the same arc and the properties of cyclic quadrilaterals.

Step 1: Determine Angle $z$

Notice that $\angle BAC$ and $\angle BDC$ subtend the same arc $BC$. According to the Inscribed Angle Theorem: $\angle BDC = \angle BAC = 49^\circ$ So, $z = 49^\circ$.

Step 2: Determine Angle $x$

To find $x$, we can use the fact that the sum of the angles in triangle $OCD$ (where $O$ is the center) must be 180°. Since $\angle OCD$ and $\angle ODC$ are given as $49^\circ$ and $23^\circ$ respectively, we have: $x = 180^\circ - 49^\circ - 23^\circ = 108^\circ$

Step 3: Determine Angle $y$

Now, angle $y$ can be found using the fact that the angle at the center is twice the angle at the circumference subtended by the same arc. Therefore: $y = \frac{1}{2} \times x = \frac{1}{2} \times 108^\circ = 54^\circ$

Summary

• $z = 49^\circ$
• $x = 108^\circ$
• $y = 54^\circ$

Would you like more detailed explanations or have any questions?

Follow-up Questions:

1. How do you determine angles in cyclic quadrilaterals?
2. What is the Inscribed Angle Theorem?
3. How do you find the center angle from a chord in a circle?
4. Can the angle at the center be less than twice the angle at the circumference?
5. What are other properties of cyclic quadrilaterals?
6. What if the triangle $OCD$ was isosceles? How would the angles change?
7. How do central angles relate to the arc length they subtend?
8. Can we solve similar problems using trigonometric identities?

Tip:

When dealing with circles, always look for relationships between the angles subtended by the same arc or chord. This often leads to straightforward solutions using known theorems.