`Delta ABC, AB = AC` and BA is produced to D such that AC = AD. Then the `angleBCD` is

Options

A. `100^@`
B. `60^@`
C. `80^@`
D. `90^@`

Answer

To solve the problem, we need to analyze the given triangle ABC where AB = AC and the line BA is extended to point D such that AC = AD. We need to find the measure of angle BCD.

### Step-by-Step Solution:

1. **Identify the Triangle and Given Information:**
- We have triangle ABC where AB = AC.
- Point D is on the extension of line BA such that AC = AD.

2. **Label the Angles:**
- Let angle ABC = x.
- Since AB = AC, by the properties of isosceles triangles, angle ACB = x as well.

3. **Determine Angle A:**
- The sum of angles in triangle ABC is 180 degrees:
\[ \text{Angle A} + \text{Angle ABC} + \text{Angle ACB} = 180 \]
- Thus, we have:
\[ \text{Angle A} + x + x = 180 \]
- This simplifies to:
\[ \text{Angle A} + 2x = 180 \]
- Therefore:
\[ \text{Angle A} = 180 - 2x \]

4. **Analyze Triangle ACD:**
- In triangle ACD, we know that AC = AD (given).
- Therefore, triangle ACD is also isosceles, which means:
\[ \text{Angle ADC} = \text{Angle ACD} = y \]
- The sum of angles in triangle ACD is also 180 degrees:
\[ \text{Angle A} + \text{Angle ADC} + \text{Angle ACD} = 180 \]
- This gives us:
\[ (180 - 2x) + y + y = 180 \]
- Simplifying this, we find:
\[ 180 - 2x + 2y = 180 \]
- Thus:
\[ 2y = 2x \quad \Rightarrow \quad y = x \]

5. **Find Angle BCD:**
- In triangle BCD, we can express the angles as follows:
\[ \text{Angle ABC} + \text{Angle BCD} + \text{Angle BDC} = 180 \]
- We know that angle BDC is equal to angle ADC, which we found to be y. Thus:
\[ x + \text{Angle BCD} + y = 180 \]
- Since y = x, we can substitute:
\[ x + \text{Angle BCD} + x = 180 \]
- This simplifies to:
\[ 2x + \text{Angle BCD} = 180 \]
- Rearranging gives:
\[ \text{Angle BCD} = 180 - 2x \]

6. **Substituting for Angle BCD:**
- We already established that angle A = 180 - 2x. Therefore:
\[ \text{Angle BCD} = 90 \]

### Conclusion:
The measure of angle BCD is 90 degrees.