Let's analyze and solve each part of the problem based on the provided information.

Given dataLet's analyze and solve each part of the problem based on the provided information.

Given

1. $QOT$ is a diameter of the circle.
2. $\angle QTP = 46^\circ$.
3. $\angle TQR = 75^\circ$.
4. $SRT = 38^\circ$.
5. $PTU$ and $RSU$ are straight lines.

We need to calculate the following angles:

Part a. $\angle RST$

Since $SRT$ is a straight line, $\angle RST = 38^\circ$ (given directly in the question).

Answer:

• $\angle RST = 38^\circ$

Reason: Given directly in the problem.

Part b. $\angle SUT$

Since $PTU$ and $RSU$ are straight lines, $\angle SUT$ is an exterior angle of triangle $SRT$. In triangles, the exterior angle is equal to the sum of the opposite interior angles. Therefore, $\angle SUT = \angle RST + \angle TQR$ Substitute the given values: $\angle SUT = 38^\circ + 75^\circ = 113^\circ$

Answer:

• $\angle SUT = 113^\circ$

Reason: Exterior angle of triangle $SRT$.

Part c. $\angle PST$

Since $PQOT$ is a cyclic quadrilateral (formed by points $P, Q, O,$ and $T$), opposite angles of a cyclic quadrilateral are supplementary. Therefore, $\angle PST + \angle QTP = 180^\circ$ Given $\angle QTP = 46^\circ$, we find $\angle PST$ by: $\angle PST = 180^\circ - 46^\circ = 134^\circ$

Answer:

• $\angle PST = 134^\circ$

Reason: Opposite angles of a cyclic quadrilateral are supplementary.

Part d. Obtuse $\angle ROT$

Since $QOT$ is a diameter, any angle subtended by a diameter at the circumference is a right angle (90°) by the angle in a semicircle property. Thus, $\angle ROT$ is obtuse and it is exactly $180^\circ - 90^\circ = 90^\circ$.

Answer:

• Obtuse $\angle ROT = 90^\circ$

Reason: Angle subtended by a diameter at the circumference of a circle.

Part e. $\angle SQT$

$\angle SQT$ can be found using the following observation: Since $PQOT$ is a cyclic quadrilateral, we have $\angle SQT$ equal to $\angle PST$ (since opposite angles are supplementary in a cyclic quadrilateral).

Thus: $\angle SQT = 46^\circ$

Answer:

• $\angle SQT = 46^\circ$

Reason: Angle in the same segment. $\angle SQT = \angle QTP$.

Let me know if you need any clarification or further details!

Follow-up Questions:

1. How do properties of cyclic quadrilaterals help in angle calculations?
2. Why is the angle subtended by a diameter always 90°?
3. How can we use straight-line properties to find unknown angles?
4. What is the importance of exterior angles in a triangle?
5. How are supplementary angles used in circle-related problems?

Tip:

Remember that angles subtended by the same chord in a circle are equal when they are on the same side of the chord. This property is useful in solving circle geometry problems.