Question

find ( mangle tqw ).
( mangle tqw = square^circ )

Explanation:

Step1: Identify straight line angles

A straight line (like \(XR\)) forms a \(180^\circ\) angle. The angles on line \(XR\) are \(67^\circ\), \(67^\circ\), and the angle between \(TQ\) and \(WQ\) (let's call it \(\angle TQW\)) along with the vertical angles? Wait, no, let's see the lines. \(XY\) and \(ZR\) seem to have vertical angles? Wait, actually, the sum of angles on a straight line is \(180^\circ\). So for the line that includes \(X\), \(Q\), and \(R\)? Wait, no, looking at the diagram, the angles around point \(Q\): the two \(67^\circ\) angles and the angle between \(TQ\) and \(WQ\) and maybe another? Wait, no, let's think about supplementary angles. Wait, the straight line (like from \(X\) through \(Q\) to... Wait, maybe the line \(XY\) and \(TW\) are vertical? No, let's calculate. The total around a point is \(360^\circ\), but for a straight line (180 degrees). So the angle between \(XQ\) and \(YQ\) is \(67^\circ\), and between \(XQ\) and \(WQ\)? Wait, no, the problem is to find \(m\angle TQW\). Let's see: the straight line (like \(XR\)) has angles \(67^\circ\) (between \(XQ\) and \(YQ\)), \(67^\circ\) (between \(XQ\) and \(ZQ\)? No, wait, maybe the line \(YQZ\) and \(TQW\) and \(XR\). Wait, the sum of angles on a straight line is \(180^\circ\). So \(67^\circ + 67^\circ + m\angle TQW + m\angle...\) No, wait, maybe the vertical angles. Wait, actually, the angle we need is \(180 - 67 - 67 = 46\). Because the three angles (67, 67, and the angle we want) are on a straight line, so \(67 + 67 + x = 180\), so \(x = 180 - 134 = 46\).

Step2: Calculate the angle

Using the straight line angle sum (\(180^\circ\)):
\(m\angle TQW = 180^\circ - 67^\circ - 67^\circ\)
\(= 180 - 134\)
\(= 46^\circ\)

Answer:

46