In this diagram, we have a regular hexagon $ABCDEF$ and two straight lines, $ACG$ and $ABH$. The angle $\angle ABH$ is given as $100^\circ$, and we are tasked with finding the value of $x$, which represents the angle $\angle GCH$.

Key observations:

1. A regular hexagon has internal angles of $120^\circ$ for each vertex.
2. Since $ABCDEF$ is a regular hexagon, the line $ACG$ can be treated as an extension, and similarly, $ABH$ is an extended straight line.

Step-by-step Solution:

1. Hexagon properties: Each interior angle of a regular hexagon is $120^\circ$, so $\angle BCA = 120^\circ$.

2. Straight line angle relationship: Since $ABH$ is a straight line, the sum of angles at point $B$ is $180^\circ$. Therefore, $\angle ABC + \angle ABH = 180^\circ.$ From the diagram, $\angle ABH = 100^\circ$, which means: $\angle ABC = 180^\circ - 100^\circ = 80^\circ.$

3. Triangle $CGH$: Now, focus on triangle $CGH$. We know one of the angles, $\angle HCG = x$, and the sum of the angles in any triangle is $180^\circ$. We already know $\angle CHG = 100^\circ$.

   To find $x$, we can use the fact that the internal angle $\angle GCH = 180^\circ - \angle BCA = 180^\circ - 120^\circ = 60^\circ$.

Thus, $x = 60^\circ$.

Final Answer:

The value of $x$ is $60^\circ$.

Would you like more detailed steps or further clarification?

Here are 5 questions to expand on this topic:

1. How can you determine the exterior angle of a regular polygon?
2. Why does the sum of angles in a straight line always equal $180^\circ$?
3. How do the properties of a regular hexagon assist in finding unknown angles?
4. Can you find the interior and exterior angles of other polygons, like pentagons or octagons?
5. How does the angle sum property of triangles help in geometry problems?

Tip: Always verify whether angles are internal or external to avoid errors in calculation.