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paragraph proof given angle A is congruent to angle D, O is the midpoint of line AD
prove triangle AOB is congruent to triangle DOC answer
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From mathematical point of view, this post is a TOTAL GIBBERISH,
since the positions of points B and C are not defined.
It is not possible to solve such a problem.
Its right place is in the garbage bin.
Below I placed a "solution" by @CPhill and marked where he makes
implicit arbitrary assumptions that are not in the problem.
We are given that angle A is congruent to angle D, and that O is the midpoint of line AD. <<<---=== Yes
Since O is the midpoint of AD, this means that AO is congruent to DO. <<<---=== Yes
We also know that angles AOB and DOC are vertical angles. <<<---=== No, we don't know it. It is not given.
It is an arbitrary assumption by @CPhill,
not supported by the problem.
So, what follows does not have a solid base/support.
Vertical angles are congruent, so angle AOB is congruent to angle DOC. <<<---=== accepted by @CPhill without any support from the problem.
We even don't know that these angles are vertical.
Therefore, we have two sides and the included angle of triangle AOB congruent <<<---=== we do not have two pairs of sides congruent.
At this point, we have only one pair of sides congruent.
to the corresponding two sides and included angle of triangle DOC.
By the Side-Angle-Side (SAS) congruence postulate, triangle AOB is congruent to triangle DOC. <<<---=== this conclusion is made without adequate base.
It is a shame how @CPhill treats this problem in his post.
In case a defective problem comes to the forum, it should be adequately
marked / declared as defective.
If a tutor wants to re-formulate the problem and to solve the modified problem,
then this modification must be done EXPLICITLY.