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The image shows a geometric figure with triangles \( \triangle ABC \) and \( \triangle ACD \), where \( AB = AC \) (indicated by the congruent marks on \( AB \) and \( AC \)) and \( CD = BD \) (indicated by the congruent marks on \( BD \) and \( CD \)). Given that \( \angle CAD = 25^\circ \), we are to find \( m\angle CAB \).

### Solution:

1. **Isosceles Triangle \( \triangle ABC \)**:
   - Since \( AB = AC \), \( \triangle ABC \) is isosceles.
   - In an isosceles triangle, the base angles are equal. Therefore, \( \angle ABC = \angle ACB = x \).

2. **Linear Pair**:
   - Angles \( \angle CAB \) and \( \angle CAD \) form a linear pair with \( \angle BAC \) in the isosceles triangle.
   - Hence, \( \angle BAC = \angle CAB + \angle CAD \).

3. **Given \( \angle CAD = 25^\circ \)**:
   - Therefore, \( \angle CAB + 25^\circ = \angle BAC \).

4. **Angle Sum Property**:
   - The sum of angles in \( \triangle ABC \) is \( 180^\circ \).
   - So, \( \angle BAC + \angle ABC + \angle ACB = 180^\circ \).
   - Since \( \angle ABC = \angle ACB = x \), we have \( \angle BAC + 2x = 180^\circ \).

5. **Using \( \angle BAC = x + 25^\circ \)**:
   - Substituting in the equation: 
     \[
     (x + 25^\circ) + 2x = 180^\circ
     \]
     \[
     3x + 25^\circ = 180^\circ
     \]
     \[
     3x = 155^\circ
     \]
     \[
     x = 51.67^\circ
     \]

6. **Finding \( \angle CAB \)**:
   - Since \( \angle BAC = x + 25^\circ \) and \( x = 51.67^\circ \):
     \[
     \angle BAC = 51.67^\circ + 25^\circ = 76.67^\circ
     \]
   - Therefore, \( \angle CAB = x = 51.67^\circ \).

Thus, \( m\angle CAB = 51.67^\circ \).

Would you like further details on any of these steps or have any other questions?

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**Related Questions:**
1. What are the properties of isosceles triangles that make solving this problem straightforward?
2. How does the concept of congruent triangles help in solving geometric problems like this?
3. Can we find the length of \( AB \) or \( AC \) if additional information is provided?
4. How does the angle sum property apply in non-Euclidean geometries?
5. What are some real-world applications of geometric principles seen here?

**Tip:** When dealing with isosceles triangles, always remember that the angles opposite the equal sides are also equal. This property is often a key step in solving related problems.

Discover how to find m∠CAB if m∠CAD = 25° in a triangle ABC where AB = AC and CD = BD. Step-by-step explanation using isosceles triangle properties and angle sum property. Suitable for high school students learning geometry.

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