Question

triangle qst is isosceles, and rt bisects ∠t. what is true about ∠qrt? choose two correct answers. ∠qrt ≅ ∠rtq ∠qrt ≅ ∠stq m∠qrt = 90° m∠qrt = m∠srt m∠qrt = 2*m∠rtq

Explanation:

Step1: Analyze the properties of isosceles triangles and angle - bisectors

Since triangle \(QST\) is isosceles and \(RT\) bisects \(\angle T\), and \(R\) is the mid - point of \(SQ\) (as \(SQ\) has two equal segments). In an isosceles triangle, the bisector of the vertex angle is also the perpendicular bisector of the base. So, \(RT\perp SQ\).

Step2: Check each option

• For \(\angle QRT\cong\angle RTQ\): There is no information to suggest that \(\triangle QRT\) is isosceles in this way.
• For \(\angle QRT\cong\angle STQ\): There is no such congruence relation based on the given information.
• For \(m\angle QRT = 90^{\circ}\): Because \(RT\) is the perpendicular bisector of \(SQ\) (from the properties of isosceles triangles and angle - bisectors), \(\angle QRT=90^{\circ}\).
• For \(m\angle QRT=m\angle SRT\): Since \(R\) is the mid - point of \(SQ\) and \(RT\perp SQ\), \(\angle QRT\) and \(\angle SRT\) are both right angles. So \(m\angle QRT = m\angle SRT\).
• For \(m\angle QRT = 2*m\angle RTQ\): There is no basis for this relationship from the given information.

Answer:

\(m\angle QRT = 90^{\circ}\), \(m\angle QRT=m\angle SRT\)