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In this diagram, we have a triangle \( ABC \) and an additional line segment \( BD \) intersecting at point \( D \) outside the triangle. Let’s solve for the unknown angle \( x \).

### Given:
- \( \angle ACB = 44^\circ \)
- \( \angle CAB = 20^\circ \)
- Line segment \( AB \) is marked as equal in length on either side of point \( D \), indicating symmetry.

### Solution Steps:
1. **Calculate \( \angle ABC \)**:
   Since \( \triangle ABC \) is a triangle, the sum of its interior angles is \( 180^\circ \).
   \[
   \angle ABC = 180^\circ - \angle ACB - \angle CAB = 180^\circ - 44^\circ - 20^\circ = 116^\circ
   \]

2. **Observe Symmetry in \( \triangle ABD \)**:
   The two segments around \( D \) on line \( AB \) indicate that angles opposite these equal sides are also equal.

3. **Solve for \( x \)**:
   Since \( \angle ABD = x \) and \( \triangle ABD \) is isosceles (two sides marked as equal), the base angles are equal. Therefore:
   \[
   2x = 180^\circ - \angle ABC = 180^\circ - 116^\circ = 64^\circ
   \]
   \[
   x = \frac{64^\circ}{2} = 32^\circ
   \]

### Answer:
The unknown angle \( x \) is \( 32^\circ \).

Would you like further details or have any questions? 

### Related Questions:
1. What is the sum of the interior angles of a triangle?
2. How do we identify isosceles triangles based on side lengths?
3. How can symmetry in geometry help solve for unknown angles?
4. Why is it important to verify angle sums in polygons?
5. What methods can be used to solve for unknowns in non-right triangles?

### Tip:
When solving for unknown angles, always look for equal side markings, as they often imply equal opposite angles.

In the given triangle ABC, find the unknown angle x, given that \( \angle ACB = 44^\circ \), \( \angle CAB = 20^\circ \), and line segment AB is marked as equal on either side of point D.

Learn how to find the unknown angle x in triangle ABC with a step-by-step solution. This problem involves key geometric concepts such as the angle sum property, isosceles triangles, and symmetry. Suitable for students in Grades 9-10.

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