In the given diagram, we are asked to determine specific angle values. The information provided includes:

• $\angle QWR = 40^\circ$
• $\angle WRT = 35^\circ$

The problem asks us to determine the following angles: a) $\angle QSR$
b) $\angle WRQ$
c) $\angle WQT$
d) $\angle QBT$

Let's analyze the situation and solve each part:

1. Analyzing the Circle and Angles:

• Chord QW = RW means that $\overline{QW}$ and $\overline{RW}$ are equal. This suggests that triangle $QWR$ is isosceles, so angles $QWR$ and $WRQ$ might be equal.
• Inscribed angles: The angle subtended by the same arc are equal, and we can use this fact to calculate angles.

Let's proceed with solving these questions:

1. $\angle QSR$: This angle is subtended by arc $QR$, the same as $\angle QWR$, so $\angle QSR = \angle QWR = 40^\circ$.

2. $\angle WRQ$: Since triangle $QWR$ is isosceles (with $QW = RW$), $\angle WRQ = \angle WQR$. Using the angle sum of a triangle: $\angle QWR + 2\cdot\angle WRQ = 180^\circ$ $40^\circ + 2\cdot\angle WRQ = 180^\circ$ Solving for $\angle WRQ$: $2\cdot\angle WRQ = 140^\circ \quad \Rightarrow \quad \angle WRQ = 70^\circ$

3. $\angle WQT$: We already have $\angle WRQ = 70^\circ$, so using this, we can calculate the remaining angles if needed.

4. $\angle QBT$: This appears to involve an angle at the circle based on tangents or other relationships.

Would you like more details on a specific angle or further clarification on the relationships?

Here are 5 follow-up questions to expand on this topic:

1. How do we determine angles formed by tangents and chords?
2. What are the properties of an isosceles triangle in circle geometry?
3. How can we apply the inscribed angle theorem to solve problems involving circles?
4. What is the relationship between angles subtended by the same arc?
5. How do you calculate the exterior angle of a triangle in circle geometry?

Tip:

In circle geometry, angles subtended by the same arc are always equal, making it easier to find missing angles based on symmetry!