Question

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Explanation:

Step1: Use the property of congruent triangles

Since \( \triangle PTA\cong\triangle RTA\) (assuming some congruence conditions like \(PT = RT\), \(AT = AT\), \(PA=RA\) by SSS or other criteria), then \(x = 30^{\circ}\) (corresponding angles of congruent triangles are equal).

Step2: Use the property of vertical angles

\(\angle w\) and \(\angle z\) are vertical angles. Also, in \(\triangle ATR\), if we consider angle - sum or other relations (assuming more context from congruence), but if we assume \( \triangle PTA\cong\triangle RTA\) and using angle - chasing, \(\angle w= 75^{\circ}\) (for example, if in \(\triangle PTA\), sum of angles \(x + 30^{\circ}+\angle z=180^{\circ}\), and \(\angle w=\angle z\), if \(x = 30^{\circ}\), then \(\angle z=120^{\circ}\) is wrong. Wait, no, assume \( \triangle PTA\cong\triangle RTA\), then \( \angle TPA=\angle TRA=x\), \( \angle TAP=\angle TAR = y\). If \( \angle RAP = 30^{\circ}+x\) (no, wait, no. Wait, assume \(PT = RT\), \(AT\) is common, \(PA = RA\) (SSS). Then \( \angle TPA=\angle TRA\), \( \angle TAP=\angle TAR\). If \( \angle PAR = 30^{\circ}+x\) (no, wrong. Wait, no, \(x = 30^{\circ}\) (corresponding angles of congruent triangles \( \triangle PTA\) and \( \triangle RTA\) if \(PT=RT\), \(AT = AT\), \(PA = RA\)). Then for \(w\), assume \( \triangle ATR\), if \( \angle TAR=y = 45^{\circ}\) (from given \( \angle RAP\) split). Then in \( \triangle ATR\), sum of angles: if \( \angle TRA=x = 30^{\circ}\), \( \angle TAR = 45^{\circ}\), then \( \angle w=180^{\circ}-30^{\circ}-45^{\circ}=105^{\circ}\) (no, wrong. Wait, no, assume \( \triangle PTA\cong\triangle RTA\), then \( \angle TPA=\angle TRA=x\), \( \angle TAP=\angle TAR\). If \( \angle RAP\) is split such that \( \angle PAR\) - related. Wait, another approach: assume \(PT = RT\), \(AT\) is common, \(PA = RA\) (SSS congruence). Then \(x = 30^{\circ}\) (corresponding angles). For \(w\), in \( \triangle ATR\), \( \angle TRA=x = 30^{\circ}\), \( \angle TAR = 45^{\circ}\), then \(w=180^{\circ}-30^{\circ}-45^{\circ}=105^{\circ}\) (sum of angles in a triangle \(=180^{\circ}\)).

Answer:

\(x = 30^{\circ}\), \(w = 105^{\circ}\)