Question

triangle qst is isosceles, and \\(\overline{rt}\\) bisects \\(\angle t\\). what is true about \\(\angle qrt\\)? select two options. \\(\square m\angle qrt = 90^\circ\\) \\(\square m\angle qrt = m\angle srt\\) \\(\square \angle qrt \cong \angle stq\\) \\(\square m\angle qrt = 2^*m\angle rtq\\) \\(\square \angle qrt \cong \angle rtq\\)

Explanation:

Step1: Identify isosceles triangle sides

Triangle QST is isosceles, so \(QS = QT\) (marked with equal segments).

Step2: Recognize angle bisector

\(\overline{RT}\) bisects \(\angle T\), so \(\angle STR = \angle QTR\).

Step3: Analyze congruent triangles

\(\triangle STR \cong \triangle QTR\) (SAS: \(ST=QT\), \(\angle STR=\angle QTR\), \(TR=TR\)). Thus, \(\angle QRT = \angle SRT\).

Step4: Check angle relationships in \(\triangle QRT\)

\(\angle QRT\) is an exterior angle of \(\triangle STR\)? No—wait, in \(\triangle QRT\), if \(\angle RTQ = x\), then since \(QS=QT\) and \(R\) divides \(QS\) equally (marked segments), \(\angle QRT = 2x = 2*m\angle RTQ\).

Answer:

B. \(m\angle QRT = m\angle SRT\), D. \(m\angle QRT = 2*m\angle RTQ\)