Question

In the adjoining figure, OA = OB and OC = OD. Show that :

1. ΔOAD ≅ ΔOBC
2. AD || CB

Answer 1

Given:

• In the figure, OA = OB and OC = OD.
• Lines AC and BD meet at O so ∠AOD and ∠BOC are vertically opposite.

To Prove:

1. ΔOAD ≅ ΔOBC.
2. AD ∥ CB.

Proof (step-wise):

1. ∠AOD = ∠BOC.

Reason: Vertically opposite angles.

2. OA = OB and OD = OC.

Reason: Given.

3. In ΔOAD and ΔOBC:

OA = OB   ...(Step 2) 

OD = OC   ...(Step 2) 

And the included angles ∠AOD = ∠BOC   ...(Step 1)

Therefore by SAS, ΔOAD ≅ ΔOBC.

Hence i is proved.

4. From the congruence (CPCT, corresponding parts of congruent triangles) we get ∠OAD = ∠OBC.

These equal angles are alternate interior angles made by the lines AD and CB with a transversal through O, so AD || CB.

Hence ii is proved.

ΔOAD ≅ ΔOBC and AD || CB, as required.