https://file.aiquickdraw.com/mathbot/file/17265916801295h1ehqr0.jpg

The question asks for the measure of \( \angle TQR \), given the triangle setup in the figure.

### Analyzing the Diagram:
- The triangle \( PQR \) is a right triangle where \( \angle PRQ = 90^\circ \).
- Point \( Q \) lies on line \( PR \) and point \( T \) lies on line \( PS \), with both having perpendicular relations forming right angles with \( PR \) and \( PS \), respectively.
- \( \angle P = 37^\circ \).

### Solution Steps:
To find \( \angle TQR \), we will rely on the properties of the triangle and angles.

1. Since \( \triangle PQR \) is a right triangle, we know the sum of angles in a triangle is \( 180^\circ \). So:
   \[
   \angle PQR = 90^\circ - \angle P = 90^\circ - 37^\circ = 53^\circ.
   \]

2. \( \angle TQR \) lies inside \( \triangle TQR \), where \( \triangle TQR \) is also a right triangle. Because \( TQ \) is perpendicular to \( PR \), this gives \( \angle TQR = \angle PQR \).

Thus, the measure of \( \angle TQR \) is:
\[
\boxed{53^\circ}.
\]

Would you like further clarification or additional details?

Here are 5 related questions you might find helpful:
1. How do we use the angle sum property in triangles?
2. How do perpendicular lines affect angle relationships?
3. Can we apply trigonometric ratios to find side lengths in right triangles?
4. What are the properties of similar triangles?
5. How can we calculate angles using complementary and supplementary angle rules?

**Tip**: Always remember, the sum of angles in a triangle is \( 180^\circ \). For right triangles, one of the angles is always \( 90^\circ \).

In the figure shown below, Q lies on PR and T lies on PS. What is the measure of angle TQR?

This question explores the properties of angles in a right triangle. Learn how to determine the measure of angle TQR using the angle sum property and right-angle relations. Suitable for students in Grades 6-9.

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