Let’s solve each question step by step using geometric principles related to circles.

Question 5:

• The angle at $A$ (the one marked "?") is the inscribed angle subtending the same arc as the angle $80^\circ$ at $C$.
• The property of inscribed angles states that they subtend half the arc they correspond to. Thus: $\text{Angle at } A = \frac{\text{Angle at } C}{2} = \frac{80^\circ}{2} = 40^\circ$

Question 6:

• The angle $42^\circ$ at $X$ is an inscribed angle subtending arc $WV$.
• By the same property: $\text{Angle at } W = \frac{\text{Arc measure of } WV}{2}$ Since the arc $WV$ corresponds to $2 \times 42^\circ$: $\text{Arc measure of } WV = 2 \times 42^\circ = 84^\circ$

Question 7:

• The given angle $35^\circ$ at $E$ is an inscribed angle subtending arc $DF$. The central angle subtending the same arc is: $\text{Angle } DPF = 2 \times 35^\circ = 70^\circ$

Question 8:

• The angles at $D, C, B$ form a triangle. The sum of angles in a triangle is $180^\circ$: $49^\circ + 70^\circ + ? = 180^\circ$ Solving for the missing angle: $? = 180^\circ - 49^\circ - 70^\circ = 61^\circ$

Final Answers:

1. Question 5: $40^\circ$
2. Question 6: $84^\circ$ (Arc measure)
3. Question 7: $70^\circ$ (Angle at $DPF$)
4. Question 8: $61^\circ$ (Angle)

Would you like detailed explanations or visual proofs for any of these?

Related Questions:

1. What is the relationship between inscribed and central angles in a circle?
2. How do we calculate arc measures from inscribed angles?
3. Can angles in a cyclic quadrilateral sum to more than $180^\circ$?
4. What is the sum of opposite angles in a cyclic quadrilateral?
5. How do you calculate the missing angle when two angles of a triangle in a circle are known?

Tip:

Always use the property of inscribed angles: they are half the measure of the central angle