https://file.aiquickdraw.com/mathbot/file/1726394462653rglv2ewl.jpg

In the given figure, we are tasked with finding the size of angle \( \text{PQS} \). Here's the key information we can extract from the image:

- The angle \( \angle SQT = 65^\circ \).
- Line \( PR \) is straight, making it a transversal intersecting triangle \( SQT \) at point \( Q \).

To solve for \( \angle PQS \), we can use the fact that:

- \( PR \) is a straight line, meaning the angles on a straight line must add up to \( 180^\circ \).
- Therefore, \( \angle PQS + \angle SQT = 180^\circ \).
  
Thus:

\[
\angle PQS = 180^\circ - 65^\circ = 115^\circ
\]

However, because of the exterior angle theorem for triangles, the exterior angle \( \angle PQS \) is equal to the sum of the interior opposite angles (which is not relevant in this case since there's no additional information about other angles of triangle \( SQT \)).

So, the correct angle should be calculated based on further steps.

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Would you like more details or have any questions on how this solution was derived?

Here are 5 related questions for practice:

1. How would you find the angle \( \angle S \) in a right-angled triangle given two sides?
2. What is the sum of the interior angles of a triangle?
3. How do you find the third angle in a triangle if you know two angles?
4. What is the exterior angle theorem in triangles?
5. What would change if the line \( PR \) was not straight?

**Tip:** When dealing with transversals and straight lines, always remember that angles on a straight line sum to \( 180^\circ \).

Determine the size of the angle PQS in figure I.

Find out how to determine the size of angle PQS using geometric principles such as the sum of angles on a straight line and the exterior angle theorem. This problem is suited for students in Grades 6-8, covering fundamental geometry concepts.

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