To solve these problems, let's analyze each question systematically:

Question 5: The arc $\overarc{CA}$ is subtended by angle $\angle BCA$ of $80^\circ$. The measure of arc $\overarc{BA}$, the angle subtended by the central angle at the center, is $?$.

Since $\overarc{CA} + \overarc{BA} = 360^\circ$ (a full circle): $\overarc{BA} = 360^\circ - 80^\circ = 280^\circ$

Question 6: The given inscribed angle $\angle XWV$ is $42^\circ$. An inscribed angle subtends an arc twice its measure. Hence: $\overarc{XV} = 2 \times 42^\circ = 84^\circ$

Question 7: The given angle $\angle DEF$ is $35^\circ$, which is an inscribed angle. Its subtended arc $\overarc{DPF}$ will be: $\overarc{DPF} = 2 \times 35^\circ = 70^\circ$

Question 8: The triangle within the circle is inscribed, so the angles $\angle BCD$, $\angle CDB$, and $\angle DBC$ sum to $180^\circ$. Given $\angle BCD = 70^\circ$ and $\angle CDB = 49^\circ$, we can calculate $\angle DBC$: $\angle DBC = 180^\circ - (70^\circ + 49^\circ) = 61^\circ$

Now, the arc subtended by $\angle DBC$ will be: $\overarc{BD} = 2 \times 61^\circ = 122^\circ$

Summary of Answers:

1. Question 5: $\overarc{BA} = 280^\circ$
2. Question 6: $\overarc{XV} = 84^\circ$
3. Question 7: $\overarc{DPF} = 70^\circ$
4. Question 8: $\overarc{BD} = 122^\circ$

Would you like a detailed breakdown of any specific step?

5 Related Questions:

1. How is the inscribed angle theorem applied to calculate arcs?
2. Why do arcs corresponding to central angles have the same measure?
3. Can you explain the relationship between inscribed angles and subtended arcs further?
4. What are other methods to find unknown angles in a circle?
5. How can we confirm the sum of arcs equals $360^\circ$?

Tip:

Always use the relationships