Question

In the figure OA = OB and OD = OC, show that
A) $$\triangle AOD\cong \triangle BOC$$
B) AD || BC

Answer 1

Hint: In this question it is given that in the above figure OA = OB and OD = OC, we have to show that $$\triangle AOD$$ and $$\triangle BOC$$ are congruent to each other and the line AD and BC are parallel. So for this we need to use SAS (side-angle-side) property of congruence and after proving congruence we can easily show that AD || BC by CPCT property.
Complete step-by-step solution:
For $$\triangle AOD$$ and $$ \triangle BOC$$,
OA = OB [Given]
$\angle AOD=\angle BOC$ [Vertically opposite angles]
OD = OC [Given]
Therefore by SAS property we can say that,
$$\triangle AOD$$ and $$\triangle BOC$$ are congruent to each other.
i.e, $$\triangle AOD\cong \triangle BOC$$
Now since the triangles are congruent then we can say by CPCT property, we can write the corresponding angles are equal, i.e, $\angle B=\angle A$.
Now as we know that AB is the transversal of the line BC and AD, and its alternate angles $\angle B$ and $\angle A$ are equal, therefore we can say that the lines AD and BC are parallel, i.e AD || BC.
Hence proved.
Note: To solve this you need to know about the CPCT property, which states that if two or more triangles which are congruent to each other then the corresponding angles and the sides of the triangles are also equal to each other.
Also we have mentioned about the vertically opposite angle, so vertically opposite angles are those angles which are opposite to each other when two lines intersect and they are equal.