1. **State the problem:** We have a quadrilateral SRQT with angles at vertices S, T, Q, and R. Given angles are \(\angle S = 100^\circ\) and \(\angle T = 61^\circ\). We need to find \(x = \angle Q\) and \(y = \angle R\). Also, \(SR = ST\) and \(RQ = QT\) indicate isosceles triangles. 2. **Use properties of isosceles triangles:** - Since \(SR = ST\), triangle SRT is isosceles with \(\angle R = \angle T\). - Since \(RQ = QT\), triangle RQT is isosceles with \(\angle Q = \angle T\). 3. **Calculate \(\angle R\):** - Given \(\angle T = 61^\circ\), and \(\angle R = \angle T\) in triangle SRT, so \(y = 61^\circ\). 4. **Calculate \(\angle Q\):** - In triangle RQT, \(\angle Q = \angle T = 61^\circ\), so \(x = 61^\circ\). 5. **Verify sum of angles in quadrilateral:** - Sum of interior angles in quadrilateral is \(360^\circ\). - Given \(\angle S = 100^\circ\), \(\angle T = 61^\circ\), \(\angle Q = 61^\circ\), \(\angle R = 61^\circ\). - Sum: \(100 + 61 + 61 + 61 = 283^\circ\), which is less than 360°, so check if \(\angle R\) is correct. 6. **Re-examine triangle SRT:** - Triangle SRT has angles \(100^\circ\) at S, \(61^\circ\) at T, so \(\angle R = 180 - 100 - 61 = 19^\circ\). - Since \(SR = ST\), \(\angle R = \angle T\) must be equal, but \(\angle T = 61^\circ\) and \(\angle R = 19^\circ\) contradicts this. 7. **Correct approach:** - Since \(SR = ST\), triangle SRT is isosceles with \(\angle R = \angle T\). - Let \(\angle R = \angle T = y\). - Sum of angles in triangle SRT: \(100 + y + y = 180\Rightarrow 2y = 80 \Rightarrow y = 40^\circ\). 8. **Similarly for triangle RQT:** - Since \(RQ = QT\), triangle RQT is isosceles with \(\angle Q = \angle T = x\). - Given \(\angle T = 61^\circ\), so \(x = 61^\circ\). 9. **Final answers:** - \(x = 61^\circ\) - \(y = 40^\circ\)