Given the problem setup, let's work through the details step by step. Here's what we know:

1. $RST$ is a straight line, implying that $\angle RST = 180^\circ$.
2. $\angle S$ in $\triangle PST = 90^\circ$, meaning that $\triangle PST$ is a right triangle.
3. $\angle P$ in $\triangle SPR = 63^\circ$.
4. $\angle S$ in $\triangle PSQ = 74^\circ$.
5. $PS = 4.3 \, \text{cm}$.
6. $ST = 5.7 \, \text{cm}$.

We need to calculate:

• a. $\angle PTS$
• b. Length of $PR$
• c. Length of $QS$

Step 1: Calculate $\angle PTS$

Since $\triangle PST$ is a right triangle and $\angle S = 90^\circ$, the remaining angle $\angle PTS$ can be calculated as: $\angle PTS = 90^\circ - \angle P$ Given $\angle P = 63^\circ$, we can find $\angle PTS$.

Step 2: Calculate Length of $PR$

In $\triangle SPR$, we know $\angle P = 63^\circ$ and the side $PS = 4.3 \, \text{cm}$. To find $PR$, we can use the sine rule or trigonometric relationships, specifically the sine function: $\sin(\angle P) = \frac{PS}{PR}$ Solving for $PR$, we have: $PR = \frac{PS}{\sin(\angle P)}$

Step 3: Calculate Length of $QS$

In $\triangle PSQ$, we know $\angle S = 74^\circ$. Since $RST$ is a straight line, $\angle PSQ + \angle QSR = 180^\circ$, and we can calculate the required lengths using trigonometric identities.

Let's calculate these values.Here are the calculated values:

• a. $\angle PTS$: $27^\circ$
• b. Length of $PR$: $4.83 \, \text{cm}$

To calculate the length of $QS$, we would need more specific details about the relationship between the points $Q$ and $S$, such as the angle $QSR$ or other dimensions of the figure.

Would you like to provide additional information, or do you have any other questions?

Here are 5 related questions:

1. How would you calculate the area of triangle $\triangle PST$?
2. Can you determine the length of $PT$ in triangle $\triangle PST$?
3. How can we use the cosine rule to solve for sides in a non-right triangle like $\triangle PSQ$?
4. What would the length of $QT$ be if you knew $\angle QST$?
5. How do different trigonometric identities apply to solving angles in quadrilaterals?

Tip: When dealing with straight-line geometries, always remember that the sum of angles on a straight line is $180^\circ$. This can help solve for unknown angles.