To solve the problem, let's analyze the triangle $\triangle ABC$ and the external angle $\angle ZABD$.

Given:

1. $\angle ABD = 100^\circ$
2. $\angle C = 35^\circ$
3. $\angle A = x^\circ$ (to find)

From the triangle angle sum property, the sum of the interior angles of a triangle is $180^\circ$. Hence, for $\triangle ABC$,

$\angle A + \angle B + \angle C = 180^\circ$

We also know that $\angle ZABD$ (external angle) equals the sum of the non-adjacent interior angles of the triangle:

$\angle ZABD = \angle A + \angle C$

Since $\angle ZABD = 100^\circ$ and $\angle C = 35^\circ$,

$\angle A + 35^\circ = 100^\circ$

Solving for $\angle A$:

$\angle A = 100^\circ - 35^\circ = 65^\circ$

Verification of triangle sum:

Using the triangle angle sum property: $\angle A + \angle B + \angle C = 65^\circ + 80^\circ + 35^\circ = 180^\circ$

Thus, the solution is consistent. The value of $\angle A$ is:

$\boxed{65^\circ}$

---

Do you need further clarification or details? Let me know!

Here are 5 related questions to expand your understanding:

1. What is the external angle theorem in geometry?
2. How do you calculate angles in a triangle when one angle is given as an exterior angle?
3. Can you prove the sum of the angles in any triangle is $180^\circ$?
4. How do you identify adjacent and opposite angles in geometry problems?
5. How can this problem be applied to non-Euclidean geometries?

Tip: Always check your calculations against the triangle angle sum property to ensure accuracy in geometry problems.