Let's go through the diagram and the questions step by step. Here's a breakdown of the provided information:

Given:

• $RS$ is a diameter of the circle centered at $O$.
• Chord $ST$ is extended to point $W$.
• Chord $SP$ extended meets the tangent $RW$ at point $V$.
• $\hat{R}_1 = 50^\circ$.

To find:

1. 9.8.1: $\hat{R}_2$ (Angle $\hat{R}_2$).
2. 9.8.2: $\hat{W}$ (Angle $\hat{W}$).
3. 9.8.3: $\hat{P}_1$ (Angle $\hat{P}_1$).
4. 9.8.4: Prove that $\hat{V}_1 = \hat{P} \hat{T} \hat{S}$.
5. 9.8.5: Prove that $WVPT$ is a cyclic quadrilateral.

Solution:

9.8.1: $\hat{R}_2$

• $RS$ is a diameter of the circle, meaning the angle subtended by $RS$ on the circumference of the circle is $90^\circ$ (angle in a semicircle).
• Since $ST$ subtends $\hat{R}_2$, and the angle at $P$ is divided by $SP$ and $PT$, the angle $\hat{R}_2$ is half of $90^\circ$.

Thus, $\hat{R}_2 = 40^\circ$.

9.8.2: $\hat{W}$

• The angle$\hat{W}$ refers to the angle formed at $W$, where $RW$ is the tangent to the circle, and $ST$ is a chord produced to $W$. By using the tangent-chord theorem, which states that the angle between a tangent and a chord is equal to the angle subtended by the chord on the opposite side of the circle, we can find $\hat{W}$.

In this case, the tangent-chord theorem tells us that:

$\hat{W} = \hat{P}_1 = 50^\circ$

9.8.3: $\hat{P}_1$

$\hat{P}_1$ is the angle subtended by chord $ST$ at point $P$ on the circumference. From the previous result:

$\hat{P}_1 = 50^\circ$

9.8.4: Prove that $\hat{V}_1 = \hat{P} \hat{T} \hat{S}$

To prove that $\hat{V}_1 = \hat{P} \hat{T} \hat{S}$, we can use the alternate segment theorem. This theorem states that the angle between a tangent and a chord is equal to the angle subtended by the chord in the alternate segment.

Since $RW$ is tangent at $R$ and chord $ST$ subtends $\hat{P} \hat{T} \hat{S}$, we have:

$\hat{V}_1 = \hat{P} \hat{T} \hat{S}$

Thus, $\hat{V}_1 = 50^\circ$, completing the proof.

9.8.5: Prove that $WVPT$ is a cyclic quadrilateral

For quadrilateral $WVPT$ to be cyclic, the opposite angles must add up to $180^\circ$.

• We know $\hat{W} = 50^\circ$ and $\hat{T}$ subtends the remaining part of the circle.
• Therefore, $\hat{T} = 180^\circ - 50^\circ = 130^\circ$.
• The other pair of opposite angles, $\hat{P}$ and $\hat{V}_1$, are both $50^\circ$, so they also satisfy the condition.

Thus, $WVPT$ is a cyclic quadrilateral.

Would you like further clarification on any part of this solution?

Here are 5 follow-up questions to explore further:

1. How do the properties of cyclic quadrilaterals relate to circle theorems?
2. Why is the tangent-chord theorem significant in solving geometric problems involving circles?
3. How can the alternate segment theorem be applied in different scenarios?
4. What is the significance of the angle subtended by a diameter being $90^\circ$?
5. How would the solution change if $RW$ were not a tangent?

Tip: In any cyclic quadrilateral, the sum of the opposite angles is always $180^\circ$.