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In the provided geometric diagram, you are asking whether \( \angle SQT = \angle UQT \) is true or false.

To determine this, let's analyze the diagram:

- The triangle \( TSU \) appears to be isosceles, as indicated by the equal markings on segments \( ST \) and \( TU \).
- The two triangles \( \triangle SQT \) and \( \triangle UQT \) share the side \( QT \).
- Additionally, the two angles \( \angle SQT \) and \( \angle UQT \) are adjacent angles around point \( Q \).

Now, since \( ST = TU \) and the triangles share a common vertex at \( Q \), it is reasonable to infer that the triangles are symmetrical with respect to line \( QT \). Therefore, the angles \( \angle SQT \) and \( \angle UQT \) are congruent, and the statement \( \angle SQT = \angle UQT \) is **true**.

Would you like further details or have any questions? Here are five related questions to expand on this:

1. How would we prove that triangles \( \triangle SQT \) and \( \triangle UQT \) are congruent?
2. What geometric properties of isosceles triangles are useful in this context?
3. Can you derive the length relationships between segments \( ST \) and \( TU \)?
4. How do adjacent angles around a point relate to each other?
5. Can we use angle bisectors in this scenario to further explain symmetry?

**Tip:** In geometry, identifying congruent triangles can often simplify problems and help to establish equal angles or sides through the congruence criteria.

Explore whether angle SQT equals angle UQT using isosceles triangle properties and congruence theorems. Analyze the symmetrical properties of the geometric figure and understand how angle relationships are derived. Suitable for Grades 8-10 students.

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