Question

In the given figure, $AB$ and $CD$ bisect each other at $O$, and $AD=BC$.

(i) State the three pairs of equal parts in $∆OAD$ and $∆OBC$.

(ii) Is $∆AOD\cong ∆BOC$? Why or why not?

(iii) Is $AD\parallel CB$? Why?

Answer 1

STEP 1 : Finding the three pairs of equal parts in $∆OAD$ and $∆OBC$.

(i)

In $∆OAD$ and $∆OBC$ as shown in figure the three pairs of equal parts are

$OA=OB$ [ Given ]

$AD=BC$ [ Given ]

$OD=OC$ [ Given ]

STEP 2 : Finding whether $∆AOD$ is congruent to $∆BOC$ or not

(ii)

Since, $OA=OB$

$AD=BC$

$OD=OC$

Therefore, by SSS congruency criteria $∆AOD\cong ∆BOC$

STEP 3 : Finding whether $AD$ is parallel to $CB$ or not

(iii)

Since, $∆AOD\cong ∆BOC$

$\Rightarrow \angle OAD=\angle OBC$ [ Corresponding Parts of Congruent Triangles are equal ]

Also, $\angle OAD$ and $\angle OBC$ are equal alternate interior angles which implies $AD$ and $CB$ are parallel.

$\therefore AD\parallel CB$

Thus, $AD\parallel CB$