Question

in the diagram, \\( \overline { q s } \\) and \\( \overline { p t } \\) are altitudes and \\( m \angle r = 57 ^ { \circ } \\). the figure is not drawn to scale. find \\( m \angle p o q \\). \\( m \angle p o q = \square ^ { \circ } \\)

Explanation:

Step1: Recall the property of the orthocenter

In a triangle, if \(QS\) and \(PT\) are altitudes, then \(O\) is the orthocenter of \(\triangle PRQ\).

Step2: Use the angle - sum property of quadrilaterals and the relationship between angles

We know that the sum of the interior angles of a quadrilateral is \(360^{\circ}\). In quadrilateral \(SORT\), \(\angle SRT = 57^{\circ}\), \(\angle OSR=\angle OTR = 90^{\circ}\) (since \(QS\perp PR\) and \(PT\perp RQ\)). So, \(\angle SOT=360^{\circ}-(90^{\circ}+90^{\circ}+57^{\circ}) = 123^{\circ}\).
Since \(\angle POQ\) and \(\angle SOT\) are vertical angles.

Step3: Apply the vertical - angles theorem

Vertical angles are equal. So, \(m\angle POQ=m\angle SOT\)

Answer:

\(123\)