ABCD is an isosceles trapezium in which `bar (AB) "||" bar(CD) and angle A = 57^(@)` . Find the value of `angle C+angleD`.

Options

A. `112^(@)`
B. `123^(@)`
C. `246^(@)`
D. `303^(@)`

Answer

To solve the problem, we need to find the value of \( \angle C + \angle D \) in the isosceles trapezium ABCD, where \( \overline{AB} \parallel \overline{CD} \) and \( \angle A = 57^\circ \).

### Step-by-Step Solution:

1. **Identify the Properties of Isosceles Trapezium:**
In an isosceles trapezium, the angles on the same side of the trapezium are equal. Therefore, we have:
\[ \angle A = \angle B \quad \text{and} \quad \angle C = \angle D \]

2. **Use the Supplementary Angle Property:**
Since \( \overline{AB} \parallel \overline{CD} \), the co-interior angles are supplementary. This means:
\[ \angle A + \angle D = 180^\circ \]
and
\[ \angle B + \angle C = 180^\circ \]

3. **Substituting the Known Angle:**
We know that \( \angle A = 57^\circ \). Therefore, we can substitute this value into the equation:
\[ 57^\circ + \angle D = 180^\circ \]

4. **Solve for Angle D:**
To find \( \angle D \), we rearrange the equation:
\[ \angle D = 180^\circ - 57^\circ = 123^\circ \]

5. **Finding Angle C:**
Since \( \angle C = \angle D \) in an isosceles trapezium, we have:
\[ \angle C = 123^\circ \]

6. **Calculate \( \angle C + \angle D \):**
Now we can find the sum of angles C and D:
\[ \angle C + \angle D = 123^\circ + 123^\circ = 246^\circ \]

### Final Answer:
\[ \angle C + \angle D = 246^\circ \]