Question

triangle qst is isosceles, and rt bisects ∠t. what is true about ∠qrt? choose two correct answers. options: m∠qrt = m∠srt, ∠qrt ≅ ∠rtq, m∠qrt = 90°, ∠qrt ≅ ∠stq, m∠qrt = 2·m∠rtq (note: ocr may have minor errors in option text)

Explanation:

Step1: Analyze Triangle QST

Triangle \( QST \) is isosceles, so \( QT = ST \) (from the marks on \( QT \) and \( ST \)) and \( QR = RS \) (from the double marks on \( QR \) and \( RS \)). \( RT \) bisects \( \angle T \), so \( \angle QTR=\angle STR \).

Step2: Check \( \triangle QRT \) and \( \triangle SRT \)

In \( \triangle QRT \) and \( \triangle SRT \):

• \( QR = RS \) (given, marked with double lines)
• \( QT = ST \) (given, marked with single lines)
• \( RT = RT \) (common side)

By SSS (Side - Side - Side) congruence criterion, \( \triangle QRT\cong\triangle SRT \). So corresponding angles \( \angle QRT \) and \( \angle SRT \) are equal, which means \( m\angle QRT = m\angle SRT \).

Now check other options:

• For \( m\angle QRT = 90^{\circ} \): There is no information to suggest that \( \angle QRT \) is a right angle. The triangle is isosceles but not necessarily right - angled.
• For \( \angle QRT\cong\angle RTQ \): In \( \triangle QRT \), \( QR = RS \) and \( QT = ST \), but there is no indication that \( QR = QT \), so we can't say \( \angle QRT=\angle RTQ \).
• For \( m\angle QRT = 2m\angle R \): There is no basis for this relationship as we don't know the measure of \( \angle R \) (assuming \( \angle R \) is \( \angle RTQ \) or other angles, but the given information doesn't support this multiple relationship).
• For \( \angle QRT\cong\angle STQ \): \( \angle STQ \) is \( \angle STR+\angle RTQ \), and \( \angle QRT \) is equal to \( \angle SRT \), not necessarily equal to \( \angle STQ \).

Answer:

• \( m\angle QRT = m\angle SRT \) (the option with \( m\angle QRT = m\angle SR T \))
• \( \angle QRT\cong\angle SRT \) (the option with \( m\angle QRT = m\angle SR T \))

(Note: Based on the analysis, the correct options are the one stating \( m\angle QRT = m\angle SRT \) and we re - check the congruence. Also, there might be a typo in the option label, but from the congruence of \( \triangle QRT \) and \( \triangle SRT \), the angle \( \angle QRT \) is equal to \( \angle SRT \))