1. **Problem statement:** We are given a quadrilateral TUVW with angles at T and V known: $\angle T = 102^\circ$ and $\angle V = 59^\circ$. Sides WT and UV are equal. We need to find: a) $\angle TUV$ b) $\angle VWT$ 2. **Key facts and formulas:** - The sum of interior angles in any quadrilateral is $360^\circ$. - Since WT = UV, triangles involving these sides may be isosceles, which helps find unknown angles. 3. **Step a) Find $\angle TUV$:** - Consider triangle TUV. - We know $\angle T = 102^\circ$ (at vertex T). - Since WT = UV, triangle WTV and triangle UVT share equal sides UV = WT. - In triangle TUV, sides TU and UV are adjacent to $\angle TUV$. - To find $\angle TUV$, note that $\angle TUV$ is the angle at vertex U between points T and V. 4. **Step b) Find $\angle VWT$:** - Consider triangle WTV. - $\angle V = 59^\circ$ is given. - Since WT = UV, triangle WTV is isosceles with WT = UV. - Use the properties of isosceles triangles to find $\angle VWT$. 5. **Calculations:** - Sum of angles in quadrilateral TUVW is $360^\circ$. - Let $\angle TUV = x$ and $\angle VWT = y$. - The other two angles are $102^\circ$ (at T) and $59^\circ$ (at V). - So, $x + y + 102 + 59 = 360$. - Simplify: $x + y + 161 = 360$. - Therefore, $x + y = 199$. 6. **Using isosceles triangle properties:** - In triangle WTV, sides WT = UV, so angles opposite these sides are equal. - $\angle VWT = \angle WTV = y$. - The angle at V is $59^\circ$. - Sum of angles in triangle WTV: $y + y + 59 = 180$. - Simplify: $2y + 59 = 180$. - $2y = 121$. - $y = 60.5^\circ$. 7. **Find $x$:** - From step 5, $x + y = 199$. - Substitute $y = 60.5$. - $x + 60.5 = 199$. - $x = 199 - 60.5 = 138.5^\circ$. **Final answers:** - a) $\angle TUV = 138.5^\circ$ - b) $\angle VWT = 60.5^\circ$