In the diagram, \(\bar{PS}\hspace{1mm} and \hspace{1mm}\bar{QT}\) are two altitudes of ?PQR. Which of the following is equal to ∠RQT?

In the diagram, \(\bar{PS}\hspace{1mm} and \hspace{1mm}\bar{QT}\) are two altitudes of ?PQR. Which of the following is equal to ∠RQT?

Answer Details

In a triangle, an altitude is a line segment drawn from a vertex perpendicular to the opposite side. In the given diagram, \(\bar{PS}\) and \(\bar{QT}\) are two altitudes of triangle PQR. By definition, the altitude \(\bar{QT}\) is perpendicular to the side \(\bar{PR}\). Therefore, ∠QTR is a right angle. Also, we know that the sum of the angles in a triangle is 180 degrees. So, we can find the measure of ∠RQT as follows: ∠RQT = 180° - ∠QTR - ∠QRT We don't know the value of either ∠QTR or ∠QRT, but we can find one of them using the fact that \(\bar{PS}\) is an altitude. Since \(\bar{PS}\) is perpendicular to \(\bar{QR}\), ∠QPS is a right angle. Therefore, ∠QTR = ∠QTP + ∠RTP Since ∠QTP and ∠RTP are both complementary to ∠QPR, which is a known angle in the diagram, we can find their values: ∠QTP = ∠RTP = 90° - ∠QPR Now we can substitute these values into our equation for ∠RQT: ∠RQT = 180° - (90° - ∠QPR) - (90° - ∠QPR) Simplifying this expression gives: ∠RQT = 2∠QPR - 90° Therefore, the answer is option (B) ?SRP, since ∠QPR and ∠SRP are corresponding angles and thus equal to each other.