ΔABC
 is right angled at B On the side AC, a point D is taken such that 
AD
=
DC
 and 
AB
=
BD
. Find the measure of 
∠
CAB
.  



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:
\[
\boxed{45^\circ}
\]

---

### 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.

ΔABC is right-angled at B. On the side AC, a point D is taken such that AD = DC and AB = BD. Find the measure of ∠CAB.

Learn how to find the measure of angle ∠CAB in a right-angled triangle where AB = BD and AD = DC, with a step-by-step breakdown of the geometry involved. The problem explores concepts of symmetry, isosceles triangles, and angle properties, ideal for students in Grades 8-10.

Math Problem Statement

ΔABC is right angled at B On the side AC, a point D is taken such that AD

DC and AB

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