If two sides of a cyclic quadrilateral are parallel , prove that the other two sides are equal

Answer

To prove that if two sides of a cyclic quadrilateral are parallel, then the other two sides are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given a cyclic quadrilateral \(ABCD\) where \(AB\) is parallel to \(DC\). We need to prove that the other two sides \(AD\) and \(BC\) are equal.

2. **Drawing the Diagram**:
Draw a cyclic quadrilateral \(ABCD\) such that \(AB\) is parallel to \(DC\). Label the points accordingly.

3. **Identifying Angles**:
Since \(AB\) is parallel to \(DC\), by the property of alternate interior angles, we have:
\[ \angle DCA = \angle CAB \]
Here, \(AC\) acts as a transversal line.

4. **Using the Property of Cyclic Quadrilaterals**:
In a cyclic quadrilateral, angles subtended by the same arc are equal. Thus, we can say:
- The angle \(DCA\) is subtended by the chord \(AD\).
- The angle \(CAB\) is subtended by the chord \(BC\).

5. **Applying the Theorem**:
Since we have established that:
\[ \angle DCA = \angle CAB \]
and both angles are subtended by the chords \(AD\) and \(BC\) respectively, we can conclude that:
\[ AD = BC \]
This follows from the theorem stating that equal angles subtended by equal chords in a circle imply that the chords are equal.

6. **Final Conclusion**:
Therefore, we have proved that if two sides of a cyclic quadrilateral are parallel, then the other two sides are equal:
\[ AD = BC \]

### Final Statement:
Hence, we conclude that the other two sides of the cyclic quadrilateral are equal.

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