Question

In the figure, \( OA, OB, OC, OD \) are drawn such that \( OB = OC \) and \( OA = OD \). Show that \( \angle A = \angle C \) and \( \angle B = \angle D \).

Hint:

When triangles share a common angle and have two pairs of equal sides, the ASA congruence condition can be directly applied to prove equality of corresponding angles.

Answer 1

Step 1: Given.
We are given that \( OB = OC \) and \( OA = OD \). We have to prove that \( \angle A = \angle C \) and \( \angle B = \angle D \).
Step 2: Construction and observation.
From the figure, consider triangles \( AOB \) and \( COD \). Given: \[ OA = OD, \quad OB = OC \] and they have a common vertical angle \( \angle AOB = \angle COD \).
Step 3: Apply the ASA congruency condition.
By the Angle-Side-Angle (ASA) criterion: \[ \triangle AOB \cong \triangle COD \]
Step 4: Corresponding parts of congruent triangles (CPCT).
Since \( \triangle AOB \cong \triangle COD \): \[ \angle A = \angle C \quad \text{and} \quad \angle B = \angle D \] Step 5: Final Conclusion.
\[ \boxed{\angle A = \angle C \text{ and } \angle B = \angle D} \]