A quadrilateral ABCD is inscribed in a circle. If AB is parallel to CD and AC = BC, then the quadrilateral must be a

Options

A. a. parallelogram
B. b. rhombus
C. c. trapezium
D. d. None of these

Answer

To solve the problem, we need to analyze the properties of the quadrilateral ABCD based on the given conditions: it is inscribed in a circle, AB is parallel to CD, and AC is equal to BC.

### Step-by-Step Solution:

1. **Understanding the Properties of the Quadrilateral**:
- Since quadrilateral ABCD is inscribed in a circle, it is a cyclic quadrilateral. This means that the opposite angles of the quadrilateral are supplementary.

2. **Identifying the Parallel Sides**:
- We know that AB is parallel to CD. This indicates that ABCD could potentially be a trapezoid since a trapezoid is defined as a quadrilateral with at least one pair of parallel sides.

3. **Using the Given Lengths**:
- We are given that AC = BC. This means that the segments from point A to C and from point B to C are equal in length. This property suggests that triangles ABC and ACD may have some symmetry.

4. **Analyzing the Implications of AC = BC**:
- Since AC = BC, triangle ABC is isosceles with AB as the base. In an isosceles triangle, the angles opposite the equal sides are equal. Therefore, angle ACB = angle ABC.

5. **Considering the Parallel Lines**:
- Since AB is parallel to CD and angles ACB and ABC are equal, by the Alternate Interior Angles Theorem, angle ADB (formed by extending line AB to point D) will also equal angle ACB.

6. **Conclusion**:
- Given that ABCD is a cyclic quadrilateral with one pair of parallel sides (AB || CD) and AC = BC, the only type of quadrilateral that satisfies all these conditions is an isosceles trapezoid. Therefore, quadrilateral ABCD must be an **isosceles trapezoid**.

### Final Answer:
The quadrilateral ABCD must be an **isosceles trapezoid**.