1. **Problem statement:** Given that AD, BE, and CF are parallel lines and IJ is parallel to GH, we are to find the value of \(\angle JIH\). The angle \(\angle GHI = 80^\circ\) and \(\angle HIG = 30^\circ\) are given. 2. **Understanding the figure and angles:** Lines AD, BE, CF are parallel, and so are IJ and GH. 3. Because \(IJ \parallel GH\), corresponding angles formed by transversal HI are equal. Hence, \(\angle JIH = \angle HIG = 30^\circ\). 4. Therefore, the measure of \(\angle JIH\) is \(30^\circ\). **Final answer:** $$\boxed{30^\circ}$$