Question

which of the following statements must be true based on the diagram below? select all that apply. (diagram is not to scale.)
answer attempt 1 out of 2
\\( \overline { s t } \\) is a segment bisector.
\\( \overline { s t } \\) is an angle bisector.
\\( s \\) is the vertex of two angles that are congruent to one another.
\\( s \\) is the midpoint of a segment in the diagram.
\\( t \\) is the midpoint of a segment in the diagram.
none of the above.

Explanation:

Step1: Check segment bisector

A segment bisector divides a segment into two equal parts. There is no information that \(ST\) bisects a segment other than \(QR\) (but we need to check mid - point for that). So, initially, we can't confirm \(ST\) is a general segment bisector.

Step2: Check angle bisector

Since the angles at \(S\) (the two angles formed by \(ST\) with \(SQ\) and \(SR\)) are marked as equal, by the definition of an angle bisector (a ray that divides an angle into two equal angles), \(ST\) is an angle bisector.

Step3: Check vertex of congruent angles

Since \(ST\) is an angle bisector (from Step 2), \(S\) is the vertex of two congruent angles (\(\angle QST\) and \(\angle RST\)).

Step4: Check mid - point of a segment for \(S\)

There is no indication in the diagram that \(S\) divides any segment into two equal parts.

Step5: Check mid - point of a segment for \(T\)

Since \(QT = TR\) (marked in the diagram), by the definition of a mid - point (a point that divides a segment into two equal parts), \(T\) is the mid - point of \(QR\).

Answer:

\(\overline{ST}\) is an angle bisector, \(S\) is the vertex of two angles that are congruent to one another, \(T\) is the midpoint of a segment in the diagram.