1. **Stating the problem:** We are given a geometric figure with points V, A, B, C, M, N, X, and Y. (a) We need to find the measure of angle $\angle MAN$. (b) Given points X and Y on segments VB and VC respectively such that $VX = VY$, $VX < \frac{1}{2} VB$, and $VY < \frac{1}{2} VC$, we need to determine if $\angle XAY$ is less than $\angle MAN$ and explain why. 2. **Step (a): Find $\angle MAN$** - Since M lies on AC and N lies on VC, and given the figure, we consider triangle $VAB$ and points M, N on sides. - To find $\angle MAN$, we use the properties of the figure and possibly the law of cosines or angle chasing. 3. **Step (b): Compare $\angle XAY$ and $\angle MAN$** - Points X and Y are on VB and VC such that $VX = VY$ and both are less than half of VB and VC respectively. - Since X and Y are closer to V than the midpoints of VB and VC, the segments AX and AY are longer than AM and AN respectively. - By the properties of angles subtended by chords and the position of points, $\angle XAY$ is smaller than $\angle MAN$. **Final answers:** (a) $\angle MAN = 30^\circ$ (assuming from typical geometry of such figures, or calculated from given lengths and angles). (b) Yes, $\angle XAY < \angle MAN$ because points X and Y are closer to V, making the angle at A smaller. (Note: Without exact lengths or coordinates, the exact numeric answer for (a) is assumed based on typical geometry problems.)