Question

sort is a parallelogram. if ∠str = 106°, which of the following statements is true? ∠rtu = 53° ∠qur = 74° ∠qst = 106° ∠rqs = 106°

Explanation:

Step1: Recall properties of parallelogram

In parallelogram SORT, opposite - angles are equal and adjacent - angles are supplementary. So, \(\angle SQR=\angle STR = 106^{\circ}\), and \(\angle QST+\angle STR = 180^{\circ}\), \(\angle QRT+\angle STR = 180^{\circ}\). Also, the diagonals of a parallelogram bisect each other.

Step2: Analyze \(\angle RTU\)

Since the diagonals of a parallelogram bisect the angles at the vertices in some cases (for rhombus, square, etc., but not in general parallelogram), and there is no information that this is a special - type parallelogram, we cannot say \(\angle RTU = 53^{\circ}\) without more information.

Step3: Analyze \(\angle QUR\)

\(\angle QUR\) and \(\angle STR\) are supplementary because they are opposite angles of the quadrilateral formed by the diagonals of the parallelogram. So, \(\angle QUR=180^{\circ}-\angle STR\). Given \(\angle STR = 106^{\circ}\), then \(\angle QUR = 180 - 106=74^{\circ}\).

Step4: Analyze \(\angle QST\)

\(\angle QST\) and \(\angle STR\) are adjacent angles in the parallelogram. So, \(\angle QST=180^{\circ}-\angle STR=180 - 106 = 74^{\circ}
eq106^{\circ}\).

Step5: Analyze \(\angle RQS\)

There is no reason for \(\angle RQS\) to be equal to \(\angle STR\) in a general parallelogram.

Answer:

\(\angle QUR = 74^{\circ}\)