1. **Problem statement:** We have two isosceles triangles TSV and UTV with given angles and side equalities. We need to find the values of angles $x$ and $y$. 2. **Given information:** - Triangle TSV is isosceles with $TS = TV$. - Triangle UTV is isosceles with $TV = UV$. - Angles given: $\angle T = 5^\circ$ in TSV, $\angle U = 70^\circ$, $\angle V = 120^\circ$ in UTV. - Unknown angles: $x = \angle S$ in TSV, $y = \angle T$ in UTV. 3. **Key properties:** - In an isosceles triangle, the angles opposite the equal sides are equal. - The sum of interior angles in any triangle is $180^\circ$. 4. **Find $x$ in triangle TSV:** - Since $TS = TV$, angles opposite these sides are equal: $\angle V = \angle S = x$. - Given $\angle T = 5^\circ$. - Sum of angles: $x + x + 5 = 180$. - Simplify: $2x + 5 = 180$. - Solve for $x$: $2x = 175 \Rightarrow x = 87.5^\circ$. 5. **Find $y$ in triangle UTV:** - Triangle UTV is isosceles with $TV = UV$, so angles opposite these sides are equal: $\angle U = \angle T = y$. - Given $\angle U = 70^\circ$, so $y = 70^\circ$. - Check sum: $y + y + 120 = 180$. - Simplify: $2y + 120 = 180$. - Solve for $y$: $2y = 60 \Rightarrow y = 30^\circ$. - But this contradicts $y = 70^\circ$ from isosceles property. 6. **Re-examine triangle UTV:** - Given $\angle V = 120^\circ$. - Sides equal: $TV = UV$ means angles opposite these sides are equal. - Angles opposite $TV$ and $UV$ are $\angle U$ and $\angle T$ respectively. - So $\angle U = \angle T = y$. - Sum of angles: $y + y + 120 = 180$. - $2y = 60 \Rightarrow y = 30^\circ$. 7. **Final answers:** - $x = 87.5^\circ$ - $y = 30^\circ$