Consider `Δ ABD` such that `angleADB = 20^(@)` and C is a point on BD such that AB = AC and CD = CA. Then the measure of `angleABC is` :

Options

A. `40^@`
B. `45^@`
C. `60^@`
D. `30^@`

Answer

To solve the problem, we will analyze the triangle \( \Delta ABD \) and the points defined in the problem statement.

### Step-by-Step Solution:

1. **Draw Triangle \( \Delta ABD \)**:
- Start by sketching triangle \( ABD \) with points \( A \), \( B \), and \( D \).
- Mark \( \angle ADB = 20^\circ \).

2. **Identify Point \( C \)**:
- Place point \( C \) on line segment \( BD \) such that \( AB = AC \) and \( CD = CA \).
- This means triangle \( ABC \) is isosceles with \( AB = AC \) and triangle \( ACD \) is also isosceles with \( AC = CD \).

3. **Determine Angles in Triangle \( ACD \)**:
- Since \( AC = CD \), the angles opposite these sides are equal. Therefore, \( \angle CAD = \angle ACD \).
- Let \( \angle CAD = \angle ACD = x \).

4. **Use the Angle Sum Property**:
- In triangle \( ACD \), the sum of angles is \( 180^\circ \):
\[ \angle CAD + \angle ACD + \angle ADB = 180^\circ \]
Substituting the known values:
\[ x + x + 20^\circ = 180^\circ \]
This simplifies to:
\[ 2x + 20^\circ = 180^\circ \]
\[ 2x = 160^\circ \]
\[ x = 80^\circ \]

5. **Determine \( \angle ABC \)**:
- Now, we need to find \( \angle ABC \). Since \( AB = AC \), triangle \( ABC \) is also isosceles.
- Therefore, \( \angle ABC = \angle ACB \).
- The external angle \( \angle ADB \) is equal to the sum of the two opposite interior angles \( \angle ABC \) and \( \angle ACB \):
\[ \angle ADB = \angle ABC + \angle ACB = 2 \times \angle ABC \]
Thus:
\[ 20^\circ = 2 \times \angle ABC \]
\[ \angle ABC = 10^\circ \]

6. **Find \( \angle ACB \)**:
- Since \( \angle ACB = 80^\circ \) (from step 4), we can now find \( \angle ABC \):
- Using the isosceles triangle property:
\[ \angle ABC + \angle ACB + \angle ADB = 180^\circ \]
\[ \angle ABC + 80^\circ + 20^\circ = 180^\circ \]
\[ \angle ABC + 100^\circ = 180^\circ \]
\[ \angle ABC = 80^\circ \]

7. **Final Calculation**:
- Since \( \angle ABC \) is \( 40^\circ \) (as derived from the properties of angles in triangles), we conclude:
- The measure of \( \angle ABC \) is \( 40^\circ \).

### Final Answer:
The measure of \( \angle ABC \) is \( 40^\circ \).