To find $z$ in the diagram, we need to use the properties of the circle and the angles subtended by the same arc.

Step 1: Identify the Relevant Angles

• The angle $\angle BAC$ subtends arc $BC$.
• Angle $\angle BDC$ also subtends the same arc $BC$.

Step 2: Apply the Inscribed Angle Theorem

According to the Inscribed Angle Theorem, angles subtended by the same arc are equal. This means: $z = \angle BDC = \angle BAC = 49^\circ$

So, $z = 49^\circ$.

Would you like further details or have any other questions?

Follow-up Questions:

1. How does the Inscribed Angle Theorem work?
2. What other theorems are useful in solving circle problems?
3. Can the value of $z$ change if the points on the circle were rearranged?
4. What if $\angle BAC$ subtended a different arc?
5. How does the angle subtended by a diameter compare with others in a circle?
6. What are some real-world applications of circle theorems?
7. How can $z$ be verified using other methods?
8. What if the circle was not drawn accurately? How would it affect the calculation?

Tip:

When solving problems involving circles, always look for arcs and the angles they subtend to apply the Inscribed Angle Theorem.