We are given a triangle $ABC$, and we are told that the angle between the altitudes from vertices $A$ and $B$ is $100^\circ$. The goal is to find the angle between the angular bisectors of angles $A$ and $B$.

Step-by-Step Solution:

1. Understanding the Setup:

   • The altitudes from points $A$ and $B$ intersect at a point (the orthocenter, $H$).
   • The angle between the altitudes from $A$ and $B$ is given as $100^\circ$.
   • We need to calculate the angle between the angle bisectors of angles $A$ and $B$, which will require us to deduce the relationships between these elements.

2. Relating the Angles: Let:

   • $\alpha = \angle A$,
   • $\beta = \angle B$,
   • $\gamma = \angle C$ (angle at vertex $C$).

   The angles of a triangle satisfy: $\alpha + \beta + \gamma = 180^\circ.$

3. Using the Altitude Angle: The angle between the altitudes from $A$ and $B$ is given as $100^\circ$. This angle can be related to the angles $\alpha$ and $\beta$ of the triangle. The angle between the altitudes is: $180^\circ - (\alpha + \beta).$ We are told this is $100^\circ$, so: $180^\circ - (\alpha + \beta) = 100^\circ,$ which simplifies to: $\alpha + \beta = 80^\circ.$

4. Finding the Angle Between Bisectors: The formula for the angle between the angle bisectors of two angles $\alpha$ and $\beta$ in a triangle is given by: $\theta = 90^\circ - \frac{\gamma}{2},$ where $\gamma$ is the angle at vertex $C$, and $\gamma = 180^\circ - (\alpha + \beta)$.

   Since $\alpha + \beta = 80^\circ$, we find: $\gamma = 180^\circ - 80^\circ = 100^\circ.$ Therefore, the angle between the bisectors is: $\theta = 90^\circ - \frac{100^\circ}{2} = 90^\circ - 50^\circ = 40^\circ.$

Final Answer:

The angle between the angular bisectors of angles $A$ and $B$ is $\boxed{40^\circ}$.

Would you like further clarification or details on any step?

Here are 5 related questions to explore this further:

1. What is the relationship between the orthocenter and angle bisectors in a triangle?
2. How would the problem change if the angle between the altitudes was different?
3. Can you generalize the angle between bisectors for any triangle?
4. How does the angle sum property influence the structure of a triangle's interior angles?
5. What are some practical applications of the angular bisector theorem in geometry?

Tip: When solving geometry problems, visualizing or drawing the figure often makes it easier to relate different elements such as angles and sides!