Question

In the figure \( OA \cdot OB = OC \cdot OD \), then prove that \( \angle A = \angle C \) and \( \angle B = \angle D \).

Hint:

When the product of two segments is equal in two triangles, they may be similar, and the corresponding angles will be equal.

Answer 1

We are given that \( OA \cdot OB = OC \cdot OD \) and we are required to prove that \( \angle A = \angle C \) and \( \angle B = \angle D \). Step 1: Use the given condition. Since \( OA \cdot OB = OC \cdot OD \), we have: \[ OA \cdot OB = OC \cdot OD. \] Step 2: Prove that \( \triangle OAB \sim \triangle OCD \). By the given condition, we can apply the angle bisector theorem or use the properties of similar triangles. Since the products of the segments are equal, we conclude that the triangles \( \triangle OAB \) and \( \triangle OCD \) are similar. Step 3: Use properties of similar triangles. For two similar triangles, the corresponding angles are equal. Therefore, we have: \[ \angle A = \angle C \quad \text{and} \quad \angle B = \angle D. \] Thus, we have proved that \( \angle A = \angle C \) and \( \angle B = \angle D \).