To analyze the properties of triangle \(QST\) with the given conditions, let's break down the given information:
- Triangle \(QST\) is isosceles with \(ST = QT\).
- Line segment \(RT\) bisects \(\angle T\).
- Segments \(SR\) and \(RQ\) are congruent, meaning \(SR = RQ\).
- Angles \(STR\) and \(RTQ\) are congruent, meaning \(\angle STR = \angle RTQ\).
From these conditions, we can derive the relevant conclusions about \(\angle QRT\):
- Since \(RT\) is the angle bisector, by the properties of angle bisectors, \(\angle TRT = \angle QRT\).
- Given that \(SR = RQ\) and \(\angle STR = \angle RTQ\), \(\triangle SRT\) and \(\triangle RQT\) can be inferred to have some congruent angles and sides.
Now let's evaluate the statements:
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Measure of angleQRT = 90°: Not necessarily true because we do not have direct information suggesting that \(\angle QRT\) is a right angle.
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Measure of angleQRT = Measure of angleSRT: This could be plausible, as \(RT\) is the angle bisector leading to \(RTQ\) being congruent to \(STR\).
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AngleQRT Is-congruent-to AngleSTQ: We can reason this out. Since \(ST = QT\) and if we've deduced that \(\angle STR\) is equal to \(\angle RTQ\), then angles opposite those equal sides (which would be \(\angle SRT\) and \(\angle QRT\)) might also be equal due to the properties of isosceles triangles.
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Measure of angleQRT = 2 * Measure of angleRTQ: This is not valid unless the angles involved have specific measures underpinning this relationship, which is not implied in the description.
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AngleQRT Is-congruent-to AngleRTQ: Given that both angles are in settings of bisectors and with the relations of triangles made by side equalities and angle equalities, this congruence can hold.
Thus, the two statements that are most likely to be true regarding \(\angle QRT\) would be:
- Measure of angleQRT = Measure of angleSRT
- AngleQRT Is-congruent-to AngleRTQ
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