1. **Problem Statement:** We are given a quadrilateral RUST with diagonals UR and WT intersecting. Given lengths US = 17 cm, VT = 6 cm, and angles \(\angle U = 42^\circ\), \(\angle T = 1.5^\circ\). We need to solve for unknown sides or angles based on the given information. 2. **Understanding the figure and given data:** The quadrilateral has diagonals UR and WT intersecting. US and VT are given as 17 cm and 6 cm respectively. Angles at U and T are 42° and 1.5° respectively. Markings indicate congruent segments on UR and US, and VT and ST. 3. **Approach:** Since the diagonals intersect and there are congruent segments, we can use properties of triangles and possibly the Law of Sines or Cosines to find unknown lengths or angles. 4. **Using the Law of Sines:** For triangle UST, if we know two sides and an angle, we can find other sides or angles. For example, in triangle UST, $$\frac{US}{\sin(\angle UST)} = \frac{ST}{\sin(\angle U)} = \frac{UT}{\sin(\angle STU)}$$ 5. **Calculate unknowns:** Since US = 17 cm and \(\angle U = 42^\circ\), and \(\angle T = 1.5^\circ\), we can find \(\angle S = 180^\circ - 42^\circ - 1.5^\circ = 136.5^\circ\). 6. **Find side ST using Law of Sines:** $$\frac{ST}{\sin(42^\circ)} = \frac{17}{\sin(136.5^\circ)}$$ $$ST = \frac{17 \times \sin(42^\circ)}{\sin(136.5^\circ)}$$ Calculate numerically: $$\sin(42^\circ) \approx 0.6691, \quad \sin(136.5^\circ) \approx 0.6947$$ $$ST \approx \frac{17 \times 0.6691}{0.6947} \approx 16.37 \text{ cm}$$ 7. **Find side UT using Law of Sines:** $$\frac{UT}{\sin(136.5^\circ)} = \frac{17}{\sin(136.5^\circ)}$$ Since this is the same ratio, UT = 17 cm (assuming triangle UST is isosceles or based on congruency markings). 8. **Summary:** Using the Law of Sines and given angles, we found side ST approximately 16.37 cm and side UT approximately 17 cm. **Final answer:** $$ST \approx 16.37 \text{ cm}, \quad UT \approx 17 \text{ cm}$$