1. **Problem statement:** We have two intersecting circles VWZ and WXYZ intersecting at points W and Z. Given: - SVT is tangent to the left circle at V. - VWX and VZY are straight lines. - $\angle TVW = 78^\circ$ - $\angle SVX = 51^\circ$ We need to find: - (a) $\angle VZW$ - (b) $\angle XYZ$ 2. **Key facts and formulas:** - The tangent to a circle is perpendicular to the radius at the point of tangency. - Angles subtended by the same chord in the same segment of a circle are equal. - The sum of angles on a straight line is $180^\circ$. 3. **Find $\angle SVW$:** Since $\angle SVX = 51^\circ$ and $\angle TVW = 78^\circ$, and $SVT$ is tangent at V, the angle between the tangent and chord VW is $\angle TVW = 78^\circ$. 4. **Calculate $\angle WVX$:** Since $VWX$ is a straight line, angles around V satisfy: $$\angle SVX + \angle WVX + \angle TVW = 180^\circ$$ Substitute known values: $$51^\circ + \angle WVX + 78^\circ = 180^\circ$$ $$\angle WVX = 180^\circ - 129^\circ = 51^\circ$$ 5. **Calculate $\angle VZW$:** In circle VWZ, chord VW subtends $\angle TVW = 78^\circ$ at the tangent and $\angle VZW$ at the circumference. By the alternate segment theorem, the angle between tangent and chord equals the angle in the alternate segment: $$\angle VZW = \angle TVW = 78^\circ$$ 6. **Calculate $\angle XYZ$:** In circle WXYZ, chord WZ subtends $\angle VZW = 78^\circ$ at point Z and $\angle XYZ$ at point Y. Angles subtended by the same chord in the same segment are equal, so: $$\angle XYZ = \angle VZW = 78^\circ$$ **Final answers:** - (a) $\angle VZW = 78^\circ$ - (b) $\angle XYZ = 78^\circ$