Given that triangle $\Delta ABC$ is right-angled at $B$, we are asked to find the measure of $\angle CAB$ under the condition that $AD = DC$ and $AB = BD$, where $D$ is a point on side $AC$.

Step-by-Step Solution:

1. Understanding the Geometry:

   • Triangle $\Delta ABC$ is a right-angled triangle with $\angle ABC = 90^\circ$.
   • The point $D$ lies on side $AC$, and it divides $AC$ into two equal parts, meaning $AD = DC$.
   • The condition $AB = BD$ implies that triangle $\Delta ABD$ is isosceles, with the two sides $AB$ and $BD$ being equal.

2. Using Triangle Properties:

   • Since triangle $\Delta ABD$ is isosceles, $\angle ABD = \angle ADB$ (the angles opposite the equal sides).
   • In triangle $\Delta ABC$, the angle sum must be 180°. Therefore, we have: $\angle ABC + \angle CAB + \angle ACB = 180^\circ$ Given that $\angle ABC = 90^\circ$, we can simplify: $\angle CAB + \angle ACB = 90^\circ$

3. Considering the Symmetry:

   • Since $AD = DC$, point $D$ is the midpoint of $AC$.
   • From the isosceles triangle $\Delta ABD$, we know that $\angle ABD = \angle ADB$. Additionally, the total angle around point $D$ is $180^\circ$, and hence $\angle ADB + \angle ABD = 180^\circ - \angle ABC = 90^\circ$.

4. Concluding the Measure of $\angle CAB$: The symmetry of the situation suggests that $\angle CAB = 45^\circ$. This is because the equalities $AD = DC$ and $AB = BD$ imply that the triangle $\Delta ABD$ splits the right angle at $B$ into two equal parts.

Thus, the measure of $\angle CAB$ is: $\boxed{45^\circ}$

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Would you like a more detailed breakdown of any of the steps?

Further questions:

1. What happens if the triangle $\Delta ABC$ is not right-angled?
2. How can the properties of isosceles triangles help in other geometric proofs?
3. What would change if $AB \neq BD$ but $AD = DC$?
4. How do you calculate angles in other types of triangles (isosceles, equilateral)?
5. Can the method of symmetry be used in non-Euclidean geometry?

Tip:

Always look for symmetry and congruence in geometric problems—they often simplify the solution process and provide insight into unknown angles or side lengths.