1. **Stating the problem:** We have a directed polygonal chain with vertices A, B, C, D, E, F, G, H and angles given at certain points: angle at B is $2x^\circ$, angle at C is $162^\circ$, and angle near G is $52^\circ$. We want to find the value of $x$. 2. **Understanding the angles:** The polygonal chain changes direction at vertices B, C, and G. The angles given are the interior angles formed by the segments meeting at these points. 3. **Using the polygonal chain angle sum property:** For a polygonal chain, the sum of the exterior angles is $360^\circ$. The exterior angle at a vertex is $180^\circ$ minus the interior angle at that vertex. 4. **Calculate the exterior angles:** - At B: interior angle is $2x^\circ$, so exterior angle is $180^\circ - 2x$. - At C: interior angle is $162^\circ$, so exterior angle is $180^\circ - 162^\circ = 18^\circ$. - At G: interior angle is $52^\circ$, so exterior angle is $180^\circ - 52^\circ = 128^\circ$. 5. **Sum of exterior angles:** Assuming these are the only vertices where the chain changes direction, the sum of exterior angles is: $$ (180 - 2x) + 18 + 128 = 360 $$ 6. **Solve for $x$:** $$ 180 - 2x + 18 + 128 = 360 $$ $$ 326 - 2x = 360 $$ $$ -2x = 360 - 326 $$ $$ -2x = 34 $$ $$ x = \frac{34}{-2} $$ $$ x = -17 $$ 7. **Interpretation:** The value $x = -17$ means the angle $2x$ is $-34^\circ$, which may indicate the direction or orientation of the angle in the polygonal chain. **Final answer:** $$ x = -17 $$